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@ -139,16 +139,16 @@ CLOSED: [2024-10-24 Thu 17:42]
* 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), springs and damper elements, is a good candidate to model the micro-station.
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), 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 use to tune the model, and to verify that a multi-body model can represent accurately the dynamics of the micro-station.
Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
Experimental modal analysis will be use to tune the model, and to verify that a multi-body model can accurately represent 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.
The tuning approach for the multi-body model based on 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 of the considering solid bodies, and the springs and dampers connecting the solid bodies.
From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considering solid bodies and the springs and dampers connecting the solid bodies.
#+begin_src latex :file modal_vibration_analysis_procedure.pdf
\begin{tikzpicture}
@ -170,19 +170,19 @@ This modal model can then be used to tune the spatial model (i.e. the multi-body
#+end_src
#+name: fig:modal_vibration_analysis_procedure
#+caption: Three models of the same structure. The goal could be to tune a spatial model (i.e. mass, stiffness and damping properties) from a response model. As an intermediate step, the modal model can prove to be very useful.
#+caption: Three models of the same structure. The goal is to tune a spatial model (i.e. mass, stiffness and damping properties) from a response model. The modal model can be used as an intermediate step.
#+RESULTS:
[[file:figs/modal_vibration_analysis_procedure.png]]
The measurement setup used to obtain the response model is presented in Section ref:sec:modal_meas_setup.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and a first analysis of the obtained signals.
The measurement setup used to obtain the response model is described in Section ref:sec:modal_meas_setup.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals.
In Section ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are computed.
These measurements are projected at the center of mass of each considered solid body to ease the further use of the results.
In Section ref:sec:modal_frf_processing, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed.
These measurements are projected at the center of mass of each considered solid body to facilitate the further use of the results.
The solid body assumption is then verified, validating the use of the multi-body model.
Finally, the modal analysis is performed in Section ref:sec:modal_analysis.
It shows how complex the micro-station dynamics is, and the necessity of having a model that represented complex dynamics.
This shows how complex the micro-station dynamics is, and the necessity of having a model representing its complex dynamics.
# #+name: tab:modal_section_matlab_code
# #+caption: Report sections and corresponding Matlab files
@ -201,10 +201,10 @@ It shows how complex the micro-station dynamics is, and the necessity of having
<<sec:modal_meas_setup>>
** Introduction :ignore:
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 assessed.
In order to perform an experimental modal analysis, a suitable measurement setup is essential.
This includes using appropriate instrumentation (presented in Section ref:ssec:modal_instrumentation) and properly preparing the structure to be measured (Section ref:ssec:modal_test_preparation).
Then, the locations of the measured motions (Section ref:ssec:modal_accelerometers) and the locations of the hammer impacts (Section ref:ssec:modal_hammer_impacts) have to be chosen carefully.
The obtained force and acceleration signals are described in Section ref:ssec:modal_measured_signals, and the quality of the measured data is assessed.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -230,10 +230,10 @@ Obtained force and acceleration signals are shown in Section ref:ssec:modal_meas
** Used Instrumentation
<<ssec:modal_instrumentation>>
Three equipment are key to perform a good modal analysis.
Three type of equipment are essential for a good modal analysis.
First, /accelerometers/ are used to measure the response of the structure.
Here, 3-axis accelerometers[fn:1] 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:&ewins00_modal chapt. 3.5.7]].
These accelerometers were glued to the micro-station using a thin layer of wax for best results [[cite:&ewins00_modal chapt. 3.5.7]].
#+name: fig:modal_analysis_instrumentation
#+caption: Instrumentation used for the modal analysis
@ -259,41 +259,41 @@ These accelerometers are glued to the micro-station using a thin layer of wax fo
#+end_subfigure
#+end_figure
Then, an /instrumented hammer/[fn:2] (figure ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled way.
Tests have been conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
The softer tip has been found to give best results as it injects more energy in the low frequency range where the coherence was low, such that the overall coherence was improved.
Then, an /instrumented hammer/[fn:2] (figure ref:fig:modal_instrumented_hammer) is used to apply forces to the structure in a controlled manner.
Tests were conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved.
Finally, an /acquisition system/[fn:3] (figure ref:fig:modal_oros) is used to acquire the injected force and the response accelerations in a synchronized way and with sufficiently low noise.
Finally, an /acquisition system/[fn:3] (figure ref:fig:modal_oros) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise.
** Structure Preparation and Test Planing
<<ssec:modal_test_preparation>>
In order to obtain meaningful results, the modal analysis of the micro-station in performed /in-situ/.
To obtain meaningful results, the modal analysis of the micro-station in performed /in-situ/.
To do so, all the micro-station stage controllers are turned "ON".
This is especially important for stages for which the stiffness is provided by local feedback control, 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.
This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
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.
The top part representing the active stabilization stage was disassembled as the active stabilization stage 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.
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 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 $n \times n$ 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.
Measuring this acrshort:frf matrix is time consuming as it requires to make $n \times n$ measurements.
However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]].
Therefore, a minimum set of $n$ frequency response functions is required.
This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ for at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
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 identified.
It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode).
In this modal analysis, it is chosen to measure the response of the structure at all considered acrshort:dof, and to excite the structure at one location in three directions in order to have some redundancy, and to ensure that all modes are properly identified.
** 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 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.
The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured.
A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
The positions of the accelerometers are visually shown on a CAD model in Figure ref:fig:modal_location_accelerometers and their precise locations with respect to a frame located at the point of interest are summarized in Table ref:tab:modal_position_accelerometers.
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 acrshort:dof can be considered per solid body.
As all key stages of the micro-station are expected to behave as solid bodies, only 6 acrshort:dof can be considered for each solid body.
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 [t]{0.60\linewidth}
@ -373,10 +373,10 @@ data2orgtable([[1:23]', 1000*acc_pos], {'Hexapod', 'Hexapod', 'Hexapod', 'Hexapo
** Hammer Impacts
<<ssec:modal_hammer_impacts>>
The chosen location of the hammer impact corresponds to the location of accelerometer number $11$ fixed to the translation stage.
The selected 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 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.
The impacts were performed in three directions, as shown in figures ref:fig:modal_impact_x, ref:fig:modal_impact_y and ref:fig:modal_impact_z.
#+name: fig:modal_hammer_impacts
#+caption: The three hammer impacts used for the modal analysis
@ -405,13 +405,13 @@ The impacts are performed in three directions, which are shown in figures ref:fi
** Force and Response signals
<<ssec:modal_measured_signals>>
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.
The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure ref:fig:modal_raw_meas.
Sharp "impacts" can be observed for the force sensor, indicating wide frequency band excitation.
For the accelerometer, a much more complex signal can be observed, indicating complex dynamics.
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).
Similar results are obtained for all the measured frequency response functions.
The "normalized" acrfull:asd of the two signals were computed and shown in Figure ref:fig:modal_asd_acc_force.
Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
Similar results were obtained for all measured frequency response functions.
#+begin_src matlab
%% Load raw data
@ -476,7 +476,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_src
#+name: fig:modal_raw_meas_asd
#+caption: Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
#+caption: Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_raw_meas}Time domain signals}
@ -494,7 +494,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_figure
The frequency response function $H_{jk}$ from the applied force $F_{k}$ to the measured acceleration $X_j$ is then computed and shown Figure ref:fig:modal_frf_acc_force.
The quality of the obtained data can be estimated using the /coherence/ function, which is shown in Figure ref:fig:modal_coh_acc_force.
The quality of the obtained data can be estimated using the /coherence/ function (Figure ref:fig:modal_coh_acc_force).
Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
#+begin_src matlab
@ -557,9 +557,9 @@ exportFig('figs/modal_coh_acc_force.pdf', 'width', 'half', 'height', 'normal');
** Introduction :ignore:
After all measurements are conducted, a $n \times p \times q$ acrlongpl:frf matrix can be 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 $\omega_{i}$
- $n = 69$: number of output measured acceleration (23 3-axis accelerometers)
- $p = 3$: number of input force excitation
- $q = 801$: number of frequency points $\omega_{i}$
For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations eqref:eq:modal_frf_matrix_raw.
@ -576,7 +576,7 @@ 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 micro-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.
Therefore, the objective of this section is to to process the Frequency Response Matrix to reduce the number of measured acrshort:dof from 69 to 36.
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.
@ -607,7 +607,7 @@ To validate this reduction of acrshort:dof and the solid body assumption, the fr
** From accelerometer DOFs to solid body DOFs
<<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).
Let us 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 acrshort:dof of the solid body expressed in the frame $\{O\}$.
#+begin_src latex :file modal_local_to_global_coordinates.pdf
@ -649,7 +649,7 @@ The goal here is to link these $4 \times 3 = 12$ measurements to the 6 acrshort:
#+RESULTS:
[[file:figs/modal_local_to_global_coordinates.png]]
The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to a reference frame $\{O\}$.
The motion of the rigid body of figure ref:fig:modal_local_to_global_coordinates can be described by its displacement $\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]$ and (small) rotations $[\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]$ with respect to the reference frame $\{O\}$.
The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\delta} p$ and $\bm{\delta \Omega}$ using equation eqref:eq:modal_compute_point_response, with $\bm{\delta\Omega}$ defined in equation eqref:eq:modal_rotation_matrix.
@ -665,7 +665,7 @@ The motion $\vec{\delta} p_{i}$ of a point $p_i$ can be computed from $\vec{\del
\end{bmatrix}
\end{equation}
Writing this in a matrix form for the four points gives eqref:eq:modal_cart_to_acc.
Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc.
\begin{equation}\label{eq:modal_cart_to_acc}
\left[\begin{array}{c}
@ -687,7 +687,7 @@ Writing this in a matrix form for the four points gives eqref:eq:modal_cart_to_a
Provided that the four sensors are properly located, the system of equation eqref:eq:modal_cart_to_acc can be solved by matrix inversion[fn:5].
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.
Therefore, having more accelerometers permits better approximation of the motion of a solid body.
\begin{equation}
\left[\begin{array}{c}
@ -708,7 +708,7 @@ Therefore, having more accelerometers permits to have a better approximation of
From the CAD model, the position of the center of mass of each considered solid body is computed (see Table ref:tab:modal_com_solid_bodies).
Then, the position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be derived.
The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
#+begin_src matlab
%% Load frequency response matrix
@ -736,12 +736,12 @@ solid_names = fields(solids);
#+end_src
#+begin_src matlab :eval no
%% Save the acceleromter positions are well as the solid bodies
%% Save the accelerometer positions are well as the solid bodies
save('mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
#+end_src
#+begin_src matlab :tangle no
%% Save the acceleromter positions are well as the solid bodies
%% Save the accelerometer positions are well as the solid bodies
save('matlab/mat/geometry.mat', 'solids', 'solid_names', 'acc_pos');
#+end_src
@ -824,9 +824,9 @@ save('matlab/mat/frf_com.mat', 'frfs_CoM');
** Verification of solid body assumption
<<ssec:modal_solid_body_assumption>>
From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered position.
In particular, the response at the location of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$.
This is what is here done to check if solid body assumption is correct in the frequency band of interest.
From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location.
In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$.
This is what is done here to check whether the solid body assumption is correct in the frequency band of interest.
#+begin_src matlab
%% Compute the FRF at the accelerometer location from the CoM reponses
@ -852,10 +852,10 @@ for exc_dir = 1:3
end
#+end_src
The comparison is made for the 4 accelerometers fixed to the micro-hexapod (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 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure ref:fig:modal_comp_acc_solid_body_frf).
The original frequency response functions and those computed from the CoM responses match well in the frequency range of interest.
Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station.
This also validates the reduction in the number of degrees of freedom from 69 (23 accelerometers with each 3 acrshort:dof) to 36 (6 solid bodies with 6 acrshort:dof).
#+begin_src matlab :exports none :results none
%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
@ -908,7 +908,7 @@ exportFig('figs/modal_comp_acc_solid_body_frf.pdf', 'width', 'full', 'height', '
#+end_src
#+name: fig:modal_comp_acc_solid_body_frf
#+caption: Comparaison of the original accelerometer response (solid curves) and reconstructed response from the solid body response (dashed curves). For accelerometers 1 to 4 corresponding to the micro-hexapod.
#+caption: Comparaison of the original accelerometer response (solid curves) and the reconstructed response from the solid body response (dashed curves). Accelerometers 1 to 4 corresponding to the micro-hexapod are shown.
#+RESULTS:
[[file:figs/modal_comp_acc_solid_body_frf.png]]
@ -918,16 +918,17 @@ exportFig('figs/modal_comp_acc_solid_body_frf.pdf', 'width', 'full', 'height', '
:END:
<<sec:modal_analysis>>
** Introduction :ignore:
The goal here is to extract the modal parameters describing the modes of station being studied, namely the natural frequencies and the modal damping (i.e. the eigenvalues) as well as the mode shapes (.i.e. the eigenvectors).
This is done from the acrshort:frf matrix previously extracted from the measurements.
In order to perform the modal parameter extraction, the order of the modal model needs to be estimated (i.e. the number of modes in the frequency band of interest).
This is done using the acrfull:mif in section ref:ssec:modal_number_of_modes.
The goal here is to extract the modal parameters describing the modes of the micro station being studied, namely, the natural frequencies and the modal damping (i.e. the eigenvalues) as well as the mode shapes (.i.e. the eigenvectors).
This is performed from the acrshort:frf matrix previously extracted from the measurements.
In order to perform the modal parameter extraction, the order of the modal model has to be estimated (i.e. the number of modes in the frequency band of interest).
This is achived using the acrfull:mif in section ref:ssec:modal_number_of_modes.
In section ref:ssec:modal_parameter_extraction, the modal parameter extraction is performed.
Graphical display of the mode shapes can be computed from the modal model, which is quite quite useful to have a physical interpretation of the modes.
The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes.
To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared with the initial measured acrshort:frf (section ref:ssec:modal_model_validity).
To validate the quality of the modal model, the full acrshort:frf matrix is computed from the modal model and compared to the initial measured acrshort:frf (section ref:ssec:modal_model_validity).
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -965,9 +966,9 @@ The complex modal indication function is defined in equation eqref:eq:modal_cmif
The acrshort:mif therefore yields to $p$ values that are also frequency dependent.
A peak in the acrshort:mif plot indicates the presence of a mode.
Repeated modes can also be detected by multiple singular values are having peaks at the same frequency.
Repeated modes can also be detected when multiple singular values have peaks at the same frequency.
The obtained acrshort:mif is shown on Figure ref:fig:modal_indication_function.
A total of 16 modes are found between 0 and $200\,\text{Hz}$.
A total of 16 modes were found between 0 and $200\,\text{Hz}$.
The obtained natural frequencies and associated modal damping are summarized in Table ref:tab:modal_obtained_modes_freqs_damps.
#+begin_src matlab
@ -1041,11 +1042,11 @@ exportFig('figs/modal_indication_function.pdf', 'width', 'wide', 'height', 'norm
** Modal parameter extraction
<<ssec:modal_parameter_extraction>>
The modal identification generally consists of curve-fitting a theoretical expression to the actual measured acrshort:frf data.
However, there are multiple level of complexity, from fitting of a single resonance, a complete curve encompassing several resonances and working on a set of many acrshort:frf plots all obtained from the same structure.
Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured acrshort:frf data.
However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many acrshort:frf plots all obtained from the same structure.
Here, the last method is used as it gives a unique and consistent model as direct output.
It takes into account the fact 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.
Here, the last method is used because it provides a unique and consistent model.
It takes into account the fact that the properties of all 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.
From the obtained modal parameters, the mode shapes are computed and can be displayed in the form of animations (three mode shapes are shown in Figure ref:fig:modal_mode_animations).
@ -1072,10 +1073,10 @@ From the obtained modal parameters, the mode shapes are computed and can be disp
#+end_subfigure
#+end_figure
These animations are quite useful to visually get a better understanding of the system dynamical behavior.
For instance, the mode shape of the first mode at $11\,\text{Hz}$ (figure ref:fig:modal_mode1_animation) indicates that there is an issue with the lower granite.
These animations are useful for visually obtaining a better understanding of the system's dynamic behavior.
For instance, the mode shape of the first mode at $11\,\text{Hz}$ (figure ref:fig:modal_mode1_animation) indicates an issue with the lower granite.
It turns out that four /Airloc Levelers/ are used to level the lower granite (figure ref:fig:modal_airloc).
These are difficult to adjust and can lead to a situation where the granite is only supported by two of them, and therefore has a low frequency "tilt mode".
These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency "tilt mode".
The levelers were then better adjusted.
#+name: fig:modal_airloc
@ -1099,10 +1100,10 @@ The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation eqref:eq:mo
** Verification of the modal model validity
<<ssec:modal_model_validity>>
In order to check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters.
Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared with the measured acrshort:frf matrix $\mathbf{H}$.
To check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters.
Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared to the measured acrshort:frf matrix $\mathbf{H}$.
In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in a matrix from as shown in equation eqref:eq:modal_eigvector_matrix.
In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in matrix from as shown in equation eqref:eq:modal_eigvector_matrix.
\begin{equation}\label{eq:modal_eigvector_matrix}
\Phi = \begin{bmatrix}
& & & & &\\
@ -1185,10 +1186,10 @@ for i = 1:size(Hsyn, 1)
end
#+end_src
The comparison between the original measured frequency response functions and the synthesized ones from the modal model is done in Figure ref:fig:modal_comp_acc_frf_modal.
Whether the obtained match can be considered good or bad is quite arbitrary.
Yet, the modal model seems to be able to represent the coupling between different nodes and different direction which is quite important in a control point of view.
This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function between a force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction.
The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure ref:fig:modal_comp_acc_frf_modal.
Whether the obtained match is good or bad is quite arbitrary.
However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective.
This can be seen in Figure ref:fig:modal_comp_acc_frf_modal_3 that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the $y$ direction to the measured acceleration at node $2$ (i.e. at the top of the micro-hexapod) in the $x$ direction.
#+begin_src matlab :exports none
acc_o = 11; dir_o = 3;
@ -1290,15 +1291,15 @@ exportFig('figs/modal_comp_acc_frf_modal_3.pdf', 'width', 'third', 'height', 'no
<<sec:modal_conclusion>>
In this study, a modal analysis of the micro-station was performed.
Thanks to adequate choice of instrumentation and proper set of measurements, high quality frequency response functions could be obtained.
The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stages architecture.
It shows lots of coupling between stages and different directions, as well as many modes.
Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained.
The obtained frequency response functions indicate that the dynamics of the micro-station are complex, which is expected from a heavy stack stage architecture.
It shows a lot of coupling between stages and different directions, and many modes.
By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage is behaving as a rigid body.
This confirms that a solid-body model can be used to properly model the micro-station.
By measuring 12 degrees of freedom on each "stage", it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
This confirms that a solid body model can be used to properly model the micro-station.
Even though lots of efforts were put in this experimental modal analysis of the micro-station, it was proven difficult to obtain an accurate modal model.
Yet, the measurements will be quite useful for tuning the parameters of the micro-station multi-body model.
Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model.
However, the measurements are useful for tuning the parameters of the micro-station multi-body model.
* Bibliography :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

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modal-analysis.tex
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@ -1,4 +1,4 @@
% Created 2024-10-24 Thu 18:44
% Created 2024-10-24 Thu 19:33
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -35,47 +35,47 @@
\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), springs and damper elements, is a good candidate to model the micro-station.
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), 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 use to tune the model, and to verify that a multi-body model can represent accurately the dynamics of the micro-station.
Although the inertia of each solid body can easily be estimated from its geometry and material density, it is more difficult to properly estimate the stiffness and damping properties of the guiding elements connecting each solid body.
Experimental modal analysis will be use to tune the model, and to verify that a multi-body model can accurately represent 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}.
The tuning approach for the multi-body model based on measurements is illustrated in Figure \ref{fig:modal_vibration_analysis_procedure}.
First, a \emph{response model} is obtained, which corresponds to a set of frequency response functions computed from experimental measurements.
From this response model, 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 of the considering solid bodies, and the springs and dampers connecting the solid bodies.
From this response model, the modal model can be computed, which consists of two matrices: one containing the natural frequencies and damping factors of the considered modes, and another describing the mode shapes.
This modal model can then be used to tune the spatial model (i.e. the multi-body model), that is, to tune the mass of the considering solid bodies and the springs and dampers connecting the solid bodies.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_vibration_analysis_procedure.png}
\caption{\label{fig:modal_vibration_analysis_procedure}Three models of the same structure. The goal could be to tune a spatial model (i.e. mass, stiffness and damping properties) from a response model. As an intermediate step, the modal model can prove to be very useful.}
\caption{\label{fig:modal_vibration_analysis_procedure}Three models of the same structure. The goal is to tune a spatial model (i.e. mass, stiffness and damping properties) from a response model. The modal model can be used as an intermediate step.}
\end{figure}
The measurement setup used to obtain the response model is presented in Section \ref{sec:modal_meas_setup}.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), the test planing, and a first analysis of the obtained signals.
The measurement setup used to obtain the response model is described in Section \ref{sec:modal_meas_setup}.
This includes the instrumentation used (i.e. instrumented hammer, accelerometers and acquisition system), test planing, and a first analysis of the obtained signals.
In Section \ref{sec:modal_frf_processing}, the obtained frequency response functions between the forces applied using the instrumented hammer and the various accelerometers fixed to the structure are computed.
These measurements are projected at the center of mass of each considered solid body to ease the further use of the results.
In Section \ref{sec:modal_frf_processing}, the obtained frequency response functions between the forces applied by the instrumented hammer and the accelerometers fixed to the structure are computed.
These measurements are projected at the center of mass of each considered solid body to facilitate the further use of the results.
The solid body assumption is then verified, validating the use of the multi-body model.
Finally, the modal analysis is performed in Section \ref{sec:modal_analysis}.
It shows how complex the micro-station dynamics is, and the necessity of having a model that represented complex dynamics.
This shows how complex the micro-station dynamics is, and the necessity of having a model representing its complex dynamics.
\chapter{Measurement Setup}
\label{sec:modal_meas_setup}
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 assessed.
In order to perform an experimental modal analysis, a suitable measurement setup is essential.
This includes using appropriate instrumentation (presented in Section \ref{ssec:modal_instrumentation}) and properly preparing the structure to be measured (Section \ref{ssec:modal_test_preparation}).
Then, the locations of the measured motions (Section \ref{ssec:modal_accelerometers}) and the locations of the hammer impacts (Section \ref{ssec:modal_hammer_impacts}) have to be chosen carefully.
The obtained force and acceleration signals are described in Section \ref{ssec:modal_measured_signals}, and the quality of the measured data is assessed.
\section{Used Instrumentation}
\label{ssec:modal_instrumentation}
Three equipment are key to perform a good modal analysis.
Three type of equipment are essential for a good modal analysis.
First, \emph{accelerometers} are used to measure the response of the structure.
Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm 5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
These accelerometers are glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
These accelerometers were glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -99,41 +99,41 @@ These accelerometers are glued to the micro-station using a thin layer of wax fo
\caption{\label{fig:modal_analysis_instrumentation}Instrumentation used for the modal analysis}
\end{figure}
Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,mV/N\) and measurement range of \(2\,kN\)} (figure \ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled way.
Tests have been conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
The softer tip has been found to give best results as it injects more energy in the low frequency range where the coherence was low, such that the overall coherence was improved.
Then, an \emph{instrumented hammer}\footnote{Kistler 9722A2000. Sensitivity of \(2.3\,mV/N\) and measurement range of \(2\,kN\)} (figure \ref{fig:modal_instrumented_hammer}) is used to apply forces to the structure in a controlled manner.
Tests were conducted to determine the most suitable hammer tip (ranging from a metallic one to a soft plastic one).
The softer tip was found to give best results as it injects more energy in the low-frequency range where the coherence was low, such that the overall coherence was improved.
Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta ADC.} (figure \ref{fig:modal_oros}) is used to acquire the injected force and the response accelerations in a synchronized way and with sufficiently low noise.
Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta ADC.} (figure \ref{fig:modal_oros}) is used to acquire the injected force and response accelerations in a synchronized manner and with sufficiently low noise.
\section{Structure Preparation and Test Planing}
\label{ssec:modal_test_preparation}
In order to obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
To obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
To do so, all the micro-station stage controllers are turned ``ON''.
This is especially important for stages for which the stiffness is provided by local feedback control, 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.
This is especially important for stages for which the stiffness is provided by local feedback control, such as the air bearing spindle, and the translation stage.
If these local feedback controls were turned OFF, this would have resulted in very low-frequency modes that were difficult to measure in practice, and it would also have led to decoupled dynamics, which would not be the case in practice.
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.
The top part representing the active stabilization stage was disassembled as the active stabilization stage 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.
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 \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 \(n \times n\) 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.
Measuring this \acrshort{frf} matrix is time consuming as it requires to make \(n \times n\) measurements.
However, due to the principle of reciprocity (\(H_{jk} = H_{kj}\)) and using the \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
Therefore, a minimum set of \(n\) frequency response functions is required.
This can be done either by measuring the response \(X_{j}\) at a fixed \acrshort{dof} \(j\) while applying forces \(F_{i}\) for at all \(n\) considered \acrshort{dof}, or by applying a force \(F_{k}\) at a fixed \acrshort{dof} \(k\) and measuring the response \(X_{i}\) for all \(n\) \acrshort{dof}.
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 identified.
It is however not advised to measure only one row or one column, as one or more modes may be missed by an unfortunate choice of force or acceleration measurement location (for instance if the force is applied at a vibration node of a particular mode).
In this modal analysis, it is chosen to measure the response of the structure at all considered \acrshort{dof}, and to excite the structure at one location in three directions in order to have some redundancy, and to ensure that all modes are properly identified.
\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 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}.
The location of the accelerometers fixed to the micro-station is essential because it defines where the dynamics is measured.
A total of 23 accelerometers were fixed to the six key stages of the micro station: the lower and upper granites, the translation stage, the tilt stage, the spindle and the micro hexapod.
The positions of the accelerometers are visually shown on a CAD model in Figure \ref{fig:modal_location_accelerometers} and their precise locations with respect to a frame located at the point of interest are summarized in Table \ref{tab:modal_position_accelerometers}.
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 \acrshort{dof} can be considered per solid body.
As all key stages of the micro-station are expected to behave as solid bodies, only 6 \acrshort{dof} can be considered for each solid body.
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}[t]{0.60\linewidth}
@ -197,10 +197,10 @@ Hexapod & 64 & -64 & -270\\
\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.
The selected 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 \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}.
The impacts were performed in three directions, as shown in figures \ref{fig:modal_impact_x}, \ref{fig:modal_impact_y} and \ref{fig:modal_impact_z}.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -227,13 +227,13 @@ The impacts are performed in three directions, which are shown in figures \ref{f
\section{Force and Response signals}
\label{ssec:modal_measured_signals}
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.
The force sensor of the instrumented hammer and the accelerometer signals are shown in the time domain in Figure \ref{fig:modal_raw_meas}.
Sharp ``impacts'' can be observed for the force sensor, indicating wide frequency band excitation.
For the accelerometer, a much more complex signal can be observed, indicating complex dynamics.
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).
Similar results are obtained for all the measured frequency response functions.
The ``normalized'' \acrfull{asd} of the two signals were computed and shown in Figure \ref{fig:modal_asd_acc_force}.
Conclusions based on the time domain signals can be clearly observed in the frequency domain (wide frequency content for the force signal and complex dynamics for the accelerometer).
Similar results were obtained for all measured frequency response functions.
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
@ -248,11 +248,11 @@ Similar results are obtained for all the measured frequency response functions.
\end{center}
\subcaption{\label{fig:modal_asd_acc_force}Amplitude Spectral Density (normalized)}
\end{subfigure}
\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the acceleromter (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Density of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
\caption{\label{fig:modal_raw_meas_asd}Raw measurement of the accelerometer (blue) and of the force sensor at the Hammer tip (red) (\subref{fig:modal_raw_meas}). Computed Amplitude Spectral Densities of the two signals (normalized) (\subref{fig:modal_asd_acc_force})}
\end{figure}
The frequency response function \(H_{jk}\) from the applied force \(F_{k}\) to the measured acceleration \(X_j\) is then computed and shown Figure \ref{fig:modal_frf_acc_force}.
The quality of the obtained data can be estimated using the \emph{coherence} function, which is shown in Figure \ref{fig:modal_coh_acc_force}.
The quality of the obtained data can be estimated using the \emph{coherence} function (Figure \ref{fig:modal_coh_acc_force}).
Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which corresponds to the frequency range of interest.
\begin{figure}[htbp]
@ -275,9 +275,9 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
\label{sec:modal_frf_processing}
After all measurements are conducted, a \(n \times p \times q\) \acrlongpl{frf} matrix can be 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
\item \(q = 801\): the number of frequency points \(\omega_{i}\)
\item \(n = 69\): number of output measured acceleration (23 3-axis accelerometers)
\item \(p = 3\): number of input force excitation
\item \(q = 801\): number of frequency points \(\omega_{i}\)
\end{itemize}
For each frequency point \(\omega_{i}\), a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations \eqref{eq:modal_frf_matrix_raw}.
@ -295,7 +295,7 @@ 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 micro-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.
Therefore, the objective of this section is to to process the Frequency Response Matrix to reduce the number of measured \acrshort{dof} from 69 to 36.
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.
@ -304,7 +304,7 @@ To validate this reduction of \acrshort{dof} and the solid body assumption, the
\section{From accelerometer DOFs to solid body DOFs}
\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).
Let us 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 \acrshort{dof} of the solid body expressed in the frame \(\{O\}\).
\begin{figure}[htbp]
@ -313,7 +313,7 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 \acrsho
\caption{\label{fig:modal_local_to_global_coordinates}Schematic of the measured motions of a solid body}
\end{figure}
The motion of the rigid body of figure \ref{fig:modal_local_to_global_coordinates} can be described by its displacement \(\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]\) and (small) rotations \([\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]\) with respect to a reference frame \(\{O\}\).
The motion of the rigid body of figure \ref{fig:modal_local_to_global_coordinates} can be described by its displacement \(\vec{\delta}p = [\delta p_x,\ \delta p_y,\ \delta p_z]\) and (small) rotations \([\delta \Omega_x,\ \delta \Omega_y,\ \delta \Omega_z]\) with respect to the reference frame \(\{O\}\).
The motion \(\vec{\delta} p_{i}\) of a point \(p_i\) can be computed from \(\vec{\delta} p\) and \(\bm{\delta \Omega}\) using equation \eqref{eq:modal_compute_point_response}, with \(\bm{\delta\Omega}\) defined in equation \eqref{eq:modal_rotation_matrix}.
@ -329,7 +329,7 @@ The motion \(\vec{\delta} p_{i}\) of a point \(p_i\) can be computed from \(\vec
\end{bmatrix}
\end{equation}
Writing this in a matrix form for the four points gives \eqref{eq:modal_cart_to_acc}.
Writing this in matrix form for the four points gives \eqref{eq:modal_cart_to_acc}.
\begin{equation}\label{eq:modal_cart_to_acc}
\left[\begin{array}{c}
@ -351,7 +351,7 @@ Writing this in a matrix form for the four points gives \eqref{eq:modal_cart_to_
Provided that the four sensors are properly located, the system of equation \eqref{eq:modal_cart_to_acc} can be solved by matrix inversion\footnote{As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.}.
The motion of the solid body expressed in a chosen frame \(\{O\}\) can be determined using equation \eqref{eq:modal_determine_global_disp}.
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.
Therefore, having more accelerometers permits better approximation of the motion of a solid body.
\begin{equation}
\left[\begin{array}{c}
@ -372,7 +372,7 @@ Therefore, having more accelerometers permits to have a better approximation of
From the CAD model, the position of the center of mass of each considered solid body is computed (see Table \ref{tab:modal_com_solid_bodies}).
Then, the position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be derived.
The position of each accelerometer with respect to the center of mass of the corresponding solid body can easily be determined.
\begin{table}[htbp]
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
@ -410,33 +410,33 @@ Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\m
\section{Verification of solid body assumption}
\label{ssec:modal_solid_body_assumption}
From the response of one solid body expressed by its 6 \acrshortpl{dof} (i.e. from \(\mathbf{H}_{\text{CoM}}\)), and using equation \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any considered position.
In particular, the response at the location of the four accelerometers can be computed and compared with the original measurements \(\mathbf{H}\).
This is what is here done to check if solid body assumption is correct in the frequency band of interest.
From the response of one solid body expressed by its 6 \acrshortpl{dof} (i.e. from \(\mathbf{H}_{\text{CoM}}\)), and using equation \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any considered location.
In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements \(\mathbf{H}\).
This is what is done here to check whether 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 (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 3 \acrshort{dof}) to 36 (6 solid bodies with 6 \acrshort{dof}).
The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure \ref{fig:modal_comp_acc_solid_body_frf}).
The original frequency response functions and those computed from the CoM responses match well in the frequency range of interest.
Similar results were obtained for the other solid bodies, indicating that the solid body assumption is valid and that a multi-body model can be used to represent the dynamics of the micro-station.
This also validates the reduction in the number of degrees of freedom from 69 (23 accelerometers with each 3 \acrshort{dof}) to 36 (6 solid bodies with 6 \acrshort{dof}).
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/modal_comp_acc_solid_body_frf.png}
\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparaison of the original accelerometer response (solid curves) and reconstructed response from the solid body response (dashed curves). For accelerometers 1 to 4 corresponding to the micro-hexapod.}
\caption{\label{fig:modal_comp_acc_solid_body_frf}Comparaison of the original accelerometer response (solid curves) and the reconstructed response from the solid body response (dashed curves). Accelerometers 1 to 4 corresponding to the micro-hexapod are shown.}
\end{figure}
\chapter{Modal Analysis}
\label{sec:modal_analysis}
The goal here is to extract the modal parameters describing the modes of station being studied, namely the natural frequencies and the modal damping (i.e. the eigenvalues) as well as the mode shapes (.i.e. the eigenvectors).
This is done from the \acrshort{frf} matrix previously extracted from the measurements.
The goal here is to extract the modal parameters describing the modes of the micro station being studied, namely, the natural frequencies and the modal damping (i.e. the eigenvalues) as well as the mode shapes (.i.e. the eigenvectors).
This is performed from the \acrshort{frf} matrix previously extracted from the measurements.
In order to perform the modal parameter extraction, the order of the modal model needs to be estimated (i.e. the number of modes in the frequency band of interest).
This is done using the \acrfull{mif} in section \ref{ssec:modal_number_of_modes}.
In order to perform the modal parameter extraction, the order of the modal model has to be estimated (i.e. the number of modes in the frequency band of interest).
This is achived using the \acrfull{mif} in section \ref{ssec:modal_number_of_modes}.
In section \ref{ssec:modal_parameter_extraction}, the modal parameter extraction is performed.
Graphical display of the mode shapes can be computed from the modal model, which is quite quite useful to have a physical interpretation of the modes.
The graphical display of the mode shapes can be computed from the modal model, which is quite quite useful for physical interpretation of the modes.
To validate the quality of the modal model, the full \acrshort{frf} matrix is computed from the modal model and compared with the initial measured \acrshort{frf} (section \ref{ssec:modal_model_validity}).
To validate the quality of the modal model, the full \acrshort{frf} matrix is computed from the modal model and compared to the initial measured \acrshort{frf} (section \ref{ssec:modal_model_validity}).
\section{Number of modes determination}
\label{ssec:modal_number_of_modes}
The \acrshort{mif} is here applied to the \(n\times p\) \acrshort{frf} matrix where \(n\) is a relatively large number of measurement DOFs (here \(n=69\)) and \(p\) is the number of excitation DOFs (here \(p=3\)).
@ -452,9 +452,9 @@ The complex modal indication function is defined in equation \eqref{eq:modal_cmi
The \acrshort{mif} therefore yields to \(p\) values that are also frequency dependent.
A peak in the \acrshort{mif} plot indicates the presence of a mode.
Repeated modes can also be detected by multiple singular values are having peaks at the same frequency.
Repeated modes can also be detected when multiple singular values have peaks at the same frequency.
The obtained \acrshort{mif} is shown on Figure \ref{fig:modal_indication_function}.
A total of 16 modes are found between 0 and \(200\,\text{Hz}\).
A total of 16 modes were found between 0 and \(200\,\text{Hz}\).
The obtained natural frequencies and associated modal damping are summarized in Table \ref{tab:modal_obtained_modes_freqs_damps}.
\begin{minipage}[t]{0.70\linewidth}
@ -495,11 +495,11 @@ Mode & Freq. [Hz] & Damp. [\%]\\
\section{Modal parameter extraction}
\label{ssec:modal_parameter_extraction}
The modal identification generally consists of curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
However, there are multiple level of complexity, from fitting of a single resonance, a complete curve encompassing several resonances and working on a set of many \acrshort{frf} plots all obtained from the same structure.
Generally, modal identification consists of curve-fitting a theoretical expression to the actual measured \acrshort{frf} data.
However, there are multiple levels of complexity, from fitting of a single resonance, to fitting a complete curve encompassing several resonances and working on a set of many \acrshort{frf} plots all obtained from the same structure.
Here, the last method is used as it gives a unique and consistent model as direct output.
It takes into account the fact 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.
Here, the last method is used because it provides a unique and consistent model.
It takes into account the fact that the properties of all 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.
From the obtained modal parameters, the mode shapes are computed and can be displayed in the form of animations (three mode shapes are shown in Figure \ref{fig:modal_mode_animations}).
@ -525,10 +525,10 @@ From the obtained modal parameters, the mode shapes are computed and can be disp
\caption{\label{fig:modal_mode_animations}Three obtained mode shape animations}
\end{figure}
These animations are quite useful to visually get a better understanding of the system dynamical behavior.
For instance, the mode shape of the first mode at \(11\,\text{Hz}\) (figure \ref{fig:modal_mode1_animation}) indicates that there is an issue with the lower granite.
These animations are useful for visually obtaining a better understanding of the system's dynamic behavior.
For instance, the mode shape of the first mode at \(11\,\text{Hz}\) (figure \ref{fig:modal_mode1_animation}) indicates an issue with the lower granite.
It turns out that four \emph{Airloc Levelers} are used to level the lower granite (figure \ref{fig:modal_airloc}).
These are difficult to adjust and can lead to a situation where the granite is only supported by two of them, and therefore has a low frequency ``tilt mode''.
These are difficult to adjust and can lead to a situation in which the granite is only supported by two of them; therefore, it has a low frequency ``tilt mode''.
The levelers were then better adjusted.
\begin{figure}[htbp]
@ -553,10 +553,10 @@ The eigenvalues \(s_r\) and \(s_r^*\) can then be computed from equation \eqref{
\section{Verification of the modal model validity}
\label{ssec:modal_model_validity}
In order to check the validity of the modal model, the complete \(n \times n\) \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) is first synthesized from the modal parameters.
Then, the elements of this \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) that were already measured can be compared with the measured \acrshort{frf} matrix \(\mathbf{H}\).
To check the validity of the modal model, the complete \(n \times n\) \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) is first synthesized from the modal parameters.
Then, the elements of this \acrshort{frf} matrix \(\mathbf{H}_{\text{syn}}\) that were already measured can be compared to the measured \acrshort{frf} matrix \(\mathbf{H}\).
In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r\) are first organized in a matrix from as shown in equation \eqref{eq:modal_eigvector_matrix}.
In order to synthesize the full \acrshort{frf} matrix, the eigenvectors \(\phi_r\) are first organized in matrix from as shown in equation \eqref{eq:modal_eigvector_matrix}.
\begin{equation}\label{eq:modal_eigvector_matrix}
\Phi = \begin{bmatrix}
& & & & &\\
@ -576,10 +576,10 @@ With \(\mathbf{H}_{\text{mod}}(\omega)\) a diagonal matrix representing the resp
\mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
\end{equation}
The comparison between the original measured frequency response functions and the synthesized ones from the modal model is done in Figure \ref{fig:modal_comp_acc_frf_modal}.
Whether the obtained match can be considered good or bad is quite arbitrary.
Yet, the modal model seems to be able to represent the coupling between different nodes and different direction which is quite important in a control point of view.
This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the frequency response function between a force applied on node 11 (i.e. on the translation stage) in the \(y\) direction to the measured acceleration at node \(2\) (i.e. at the top of the micro-hexapod) in the \(x\) direction.
The comparison between the original measured frequency response functions and those synthesized ones from the modal model is presented in Figure \ref{fig:modal_comp_acc_frf_modal}.
Whether the obtained match is good or bad is quite arbitrary.
However, the modal model seems to be able to represent the coupling between different nodes and different directions, which is quite important from a control perspective.
This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the frequency response function from the force applied on node 11 (i.e. on the translation stage) in the \(y\) direction to the measured acceleration at node \(2\) (i.e. at the top of the micro-hexapod) in the \(x\) direction.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -607,15 +607,15 @@ This can be seen in Figure \ref{fig:modal_comp_acc_frf_modal_3} that shows the f
\label{sec:modal_conclusion}
In this study, a modal analysis of the micro-station was performed.
Thanks to adequate choice of instrumentation and proper set of measurements, high quality frequency response functions could be obtained.
The obtained frequency response functions indicate that the dynamics of the micro-station is complex, which is expected from a heavy stack stages architecture.
It shows lots of coupling between stages and different directions, as well as many modes.
Thanks to an adequate choice of instrumentation and proper set of measurements, high quality frequency response functions can be obtained.
The obtained frequency response functions indicate that the dynamics of the micro-station are complex, which is expected from a heavy stack stage architecture.
It shows a lot of coupling between stages and different directions, and many modes.
By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage is behaving as a rigid body.
This confirms that a solid-body model can be used to properly model the micro-station.
By measuring 12 degrees of freedom on each ``stage'', it could be verified that in the frequency range of interest, each stage behaved as a rigid body.
This confirms that a solid body model can be used to properly model the micro-station.
Even though lots of efforts were put in this experimental modal analysis of the micro-station, it was proven difficult to obtain an accurate modal model.
Yet, the measurements will be quite useful for tuning the parameters of the micro-station multi-body model.
Although a lot of effort was put into this experimental modal analysis of the micro-station, it was difficult to obtain an accurate modal model.
However, the measurements are useful for tuning the parameters of the micro-station multi-body model.
\printbibliography[heading=bibintoc,title={Bibliography}]