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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.
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 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.
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.
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.
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 the developed more complex multi-body model.
# #+name: tab:modal_section_matlab_code
# #+caption: Report sections and corresponding Matlab files
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.
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.
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.
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".
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.
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.
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.
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.
The location of the accelerometers fixed to the micro-station is essential as it defines where the dynamics is identified.
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).
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]].
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.
The force sensor 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.
The "normalized" amplitude spectral density 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).
#+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})
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.
#+caption: Computed frequency response function from the applied force $F_{k}$ and the measured response $X_{j}$ (\subref{fig:modal_frf_acc_force}) as well as computed coherence (\subref{fig:modal_coh_acc_force})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:modal_frf_acc_force} Frequency Response Function}
All measurements where conducted and a $n \times p \times q$ Frequency Response Functions 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 $\omega_{i}$
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.
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 DoFs from 69 to 36.
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 $69 \times 3 \times 801$ 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.
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.
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.
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.
#+caption: Comparaison of measured frequency response function for in the X directions for accelerometers aligned along X. Amplitude is in $\frac{m/s^2}{N}$. Accelerometer number is shown in the legend.
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 $\{O\}$.
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.
Therefore, having more accelerometers permits to have a better approximation of the motion of the solid body.
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 $\bm{H}$.
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.
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).
#+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.
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.
In order to do the modal parameter extraction, we first have to estimate the order of the modal model we want to obtain.
This corresponds to how many modes are present in the frequency band of interest.
In section 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*.
However, there are multiple level of complexity:
- works on a part of a single FRF curve
- works on a complete curve encompassing several resonances
- works on a set of many FRF plots all obtained from the same structure
The third method is the most complex but gives better results. This is the one we will use in section 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).
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.
**** Singular Value Decomposition - Modal Indication Function
The Mode Indicator Functions are usually used on $n\times p$ FRF matrix where $n$ is a relatively large number of measurement DOFs and $p$ is the number of excitation DOFs, typically 3 or 4.
In these methods, the frequency dependent FRF matrix is subjected to a singular value decomposition analysis which thus yields a small number (3 or 4) of singular values, these also being frequency dependent.
These methods are used to *determine the number of modes* present in a given frequency range, to *identify repeated natural frequencies* and to pre-process the FRF data prior to modal analysis.
From the documentation of the modal software:
#+begin_quote
The MIF consist of the singular values of the Frequency response function matrix. The number of MIFs equals the number of excitations.
By the powerful singular value decomposition, the real signal space is separated from the noise space. Therefore, the MIFs exhibit the modes effectively.
A peak in the MIFs plot usually indicate the existence of a structural mode, and two peaks at the same frequency point means the existence of two repeated modes.
Moreover, the magnitude of the MIFs implies the strength of the a mode.
#+end_quote
#+begin_important
The *Complex Mode Indicator Function* is defined simply by the SVD of the FRF (sub) matrix:
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.
Such method also have the advantage of producing a *unique and consistent model* as direct output.
In order to apply this method, we select the frequency range of interest and we give an estimate of how many modes are present.
Then, it shows a stabilization charts, such as the one shown on figure ref:fig:modal_stabilization_chart, where we have to manually select which modes to take into account in the modal model.
We can learn quite a lot from these mode shape animations.
For instance, the mode shape of the first mode at 11Hz (figure ref:fig:modal_mode1_animation) seems to indicate that this corresponds to a suspension mode.
This could be due to the 4 Airloc Levelers that are used for the granite (figure ref:fig:modal_airloc).
#+name: fig:modal_airloc
#+caption: AirLoc used for the granite (2120-KSKC)
#+attr_latex: :width 0.6\linewidth
[[file:figs/modal_airlock_picture.jpg]]
They are probably *not well leveled*, so the granite is supported only by two Airloc.
** Verify the validity of the Modal Model
<<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
From the modal model, we want to synthesize the Frequency Response Functions that has been used to build the modal model.
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:
Scaling constant for a complex mode. It has the same properties as modal mass for normal modes (undamped or proportionally damped cases). Assuming
- $\psi_{pr}$ = Modal coefficient for measured degree of freedom p and mode r
- $\psi_{qr}$ = Modal coefficient for measured degree of freedom q and mode r
- $A_{pqr}$ = Residue for measured degree of freedom p, measured degree of q and mode r
- $M_{Ar}$ = Modal A of mode r
Then
\[ A_{pqr} = \frac{\psi_{pr}\psi_{qr}}{M_{Ar}} \]
*Modal B*
Scaling constant for a complex mode. It has the same properties as modal stiffness for normal modes (undamped or proportionally damped cases). Assuming
The comparison between the original measured frequency response function and the synthesized one from the modal model is done in Figure ref:fig:modal_comp_acc_frf_modal.
# Check if I have the compliance of the micro-station, could be useful to compare and verify the quality of the model
# TODO - say it will be useful when validating the multi-body model
# #+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
# #+attr_latex: :width \linewidth
# [[file:img/modal_software/from11xto1x.jpg]]
# #+begin_warning
# The synthesize frf from the modal software do not match the synthesize frf computed here.
# It is possible that it uses residues that are not exported? Nothing is said about that in the documentation.
# #+end_warning
Frequency response functions that has not been measured can be synthesized as shown in Figure ref:fig:modal_synthesized_frf.
