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Rework "frequency analysis" section

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Thomas Dehaeze 2024-06-26 17:58:36 +02:00
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@ -97,6 +97,8 @@
* Notes :noexport:
Prefix is =modal=
** TODO [#C] Section ref:ssec:modal_solid_body_first_check can probably be removed
As the solid body assumption is already verified in section ref:ssec:modal_solid_body_assumption.
* Introduction :ignore:
@ -366,69 +368,6 @@ This excitation point and the three considered directions allows to properly exc
#+end_subfigure
#+end_figure
** Signal Processing :noexport:
<<ssec:modal_signal_processing>>
The measurements are averaged 10 times corresponding to 10 hammer impacts in order to reduce the effect of random noise.
Windowing is also used on the force and response signals.
A boxcar window (figure ref:fig:modal_windowing_force_signal) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.
The parameters are:
- *Start*: $3\%$
- *Stop*: $7\%$
#+begin_src matlab :exports none :results none
%% Boxcar window used for the force signal
figure;
plot(100*[0, 0.03, 0.03, 0.07, 0.07, 1], [0, 0, 1, 1, 0, 0]);
xlabel('Time [\%]'); ylabel('Amplitude');
xlim([0, 100]); ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_windowing_force_signal.pdf', 'width', 'normal', 'height', 'normal');
#+end_src
#+name: fig:modal_windowing_force_signal
#+caption: Boxcar window used for the force signal
#+RESULTS:
[[file:figs/modal_windowing_force_signal.png]]
An exponential window (figure ref:fig:modal_windowing_acc_signal) is used for the response signal as we are measuring transient signals and most of the information is located at the beginning of the signal.
The parameters are:
- FlatTop:
- *Start*: $3\%$
- *Stop*: $2.96\%$
- Decreasing point:
- *X*: $60.4\%$
- *Y*: $14.7\%$
#+begin_src matlab :exports none :results none
%% Exponential window used for acceleration signal
x0 = 0.296;
xd = 0.604;
yd = 0.147;
alpha = log(yd)/(x0 - xd);
t = x0:0.01:1.01;
y = exp(-alpha*(t-x0));
figure;
plot(100*[0, 0.03, 0.03, x0, t], [0, 0, 1, 1, y]);
xlabel('Time [\%]'); ylabel('Amplitude');
xlim([0, 100]); ylim([0, 1]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/modal_windowing_acc_signal.pdf', 'width', 'normal', 'height', 'normal');
#+end_src
#+name: fig:modal_windowing_acc_signal
#+caption: Exponential window used for acceleration signal
#+RESULTS:
[[file:figs/modal_windowing_acc_signal.png]]
** Force and Response signals
<<ssec:modal_measured_signals>>
@ -519,7 +458,7 @@ exportFig('figs/modal_asd_acc_force.pdf', 'width', 'half', 'height', 'normal');
#+end_subfigure
#+end_figure
The frequency response function 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 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.
Good coherence is obtained from $20\,\text{Hz}$ to $200\,\text{Hz}$ which corresponds to the frequency range of interest.
@ -581,15 +520,16 @@ exportFig('figs/modal_coh_acc_force.pdf', 'width', 'half', 'height', 'normal');
:END:
<<sec:modal_frf_processing>>
** Introduction :ignore:
The measurements have been conducted and a $n \times p \times q$ Frequency Response Functions Matrix has been computed with:
- $n$: the number of output measured accelerations: $23 \times 3 = 69$ (23 accelerometers measuring 3 directions each)
- $p$: the number of input force excitations: $3$
- $q$: the number of frequency points $\omega_i$
Thus, the FRF matrix is an $69 \times 3 \times 801$ matrix.
For each frequency point $\omega_i$, a 2D matrix is obtained that links the 3 force inputs to the 69 output accelerations:
\begin{equation}
\text{FRF}(\omega_i) = \begin{bmatrix}
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.
\begin{equation}\label{eq:modal_frf_matrix_raw}
\mathbf{H}(\omega_i) = \begin{bmatrix}
\frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\
\frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\
\frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\
@ -604,17 +544,12 @@ 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 measures stages have to behave as rigid bodies in the frequency band of interest.
This
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).
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 go from the accelerometers DoFs to the solid body 6 DoFs (three translations and three rotations), some computations have to be performed.
This is explained in Section ref:ssec:modal_acc_to_solid_dof.
Finally, the $69 \times 3 \times 801$ frequency response matrix is reduced to a $36 \times 3 \times 801$ frequency response matrix where the motion of each solid body is expressed with respect to the CoM of the solid body (Section ref:ssec:modal_frf_com).
To validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section ref:ssec:modal_solid_body_assumption.
To further validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section ref:ssec:modal_solid_body_assumption.
** Matlab Init :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
@ -639,8 +574,9 @@ To validate this reduction of DoF and the solid body assumption, the frequency r
** First verification of the solid body assumption
<<ssec:modal_solid_body_first_check>>
In this section, it is shown that two accelerometers fixed to a rigid body at positions $\vec{p}_1$ and $\vec{p}_2$ in such a way that $\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}$ will measure the same value in the $\vec{x}$ direction.
Such situation is illustrated on Figure ref:fig:modal_aligned_accelerometers and is the case for many accelerometers as shown in Figure ref:fig:modal_location_accelerometers.
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.
#+begin_src latex :file modal_aligned_accelerometers.pdf :results file raw silent
\newcommand\irregularcircle[2]{% radius, irregularity
@ -667,8 +603,8 @@ Such situation is illustrated on Figure ref:fig:modal_aligned_accelerometers and
\coordinate[] (p1) at (-1.5, 1.5);
\coordinate[] (p2) at ( 1.5, 1.5);
\draw[->] (p1)node[]{$\bullet$}node[above]{$p_1$} -- ++(1, 0)node[above]{$\delta p_{x1}$};
\draw[->] (p2)node[]{$\bullet$}node[above]{$p_2$} -- ++(1, 0)node[above]{$\delta p_{x2}$};
\draw[->] (p1)node[]{$\bullet$}node[above]{$\vec{p}_1$} -- ++(1, 0)node[above]{$\delta p_{x1}$};
\draw[->] (p2)node[]{$\bullet$}node[above]{$\vec{p}_2$} -- ++(1, 0)node[above]{$\delta p_{x2}$};
\draw[dashed] ($(p1)+(-1, 0)$) -- ($(p2)+(2, 0)$);
\end{tikzpicture}
@ -679,39 +615,59 @@ Such situation is illustrated on Figure ref:fig:modal_aligned_accelerometers and
#+RESULTS:
[[file:figs/modal_aligned_accelerometers.png]]
The motion of the rigid body of figure ref:fig:modal_aligned_accelerometers can be defined by its displacement $\delta \vec{p}$ and rotation $\vec{\Omega}$ with respect to a reference frame $\{O\}$.
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 motions at points $1$ and $2$ are:
\begin{align*}
\delta \vec{p}_1 &= \delta \vec{p} + \vec{\Omega} \times \vec{p}_1 \\
\delta \vec{p}_2 &= \delta \vec{p} + \vec{\Omega} \times \vec{p}_2
\end{align*}
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.
Taking only the $x$ direction:
\begin{align*}
\delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
\delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
\end{align*}
\begin{subequations}\label{eq:modal_p1_p2_motion}
\begin{align}
\vec{\delta} p_{1} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{1} \\
\vec{\delta} p_{2} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{2}
\end{align}
\end{subequations}
However, we have $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$ because of the co-linearity of the two sensors in the $x$ direction, and thus we obtain:
\begin{equation}
\begin{equation}\label{eq:modal_rotation_matrix}
\bm{\delta\Omega} = \begin{bmatrix}
0 & -\delta\Omega_z & \delta\Omega_y \\
\delta\Omega_z & 0 & -\delta\Omega_x \\
-\delta\Omega_y & \delta\Omega_x & 0
\end{bmatrix}
\end{equation}
Considering only the $x$ direction, equation eqref:eq:modal_p1_p2_x_motion is obtained.
\begin{subequations}\label{eq:modal_p1_p2_x_motion}
\begin{align}
\delta p_{x1} &= \delta p_x + \delta \Omega_y p_{z1} - \delta \Omega_z p_{y1} \\
\delta p_{x2} &= \delta p_x + \delta \Omega_y p_{z2} - \delta \Omega_z p_{y2}
\end{align}
\end{subequations}
Because the two sensors are co-linearity in the $x$ direction, $p_{1y} = p_{2y}$ and $p_{1z} = p_{2z}$, and eqref:eq:modal_colinear_sensors_equal is obtained.
\begin{equation}\label{eq:modal_colinear_sensors_equal}
\boxed{\delta p_{x1} = \delta p_{x2}}
\end{equation}
It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction "linking" the two sensors.
It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction "in line" the two sensors.
Such property can be used to verify that the considered stages are indeed behaving as rigid body in the frequency band of interest.
From Table ref:tab:modal_position_accelerometers, we can identify which pair of accelerometers are aligned in the X and Y directions.
From Table ref:tab:modal_position_accelerometers, the pairs of accelerometers that aligned in the X and Y directions can be identified.
The response in the X direction of pairs of sensors aligned in the X direction are compared in Figure ref:fig:modal_solid_body_comp_x_dir.
Good match is observed up to 200Hz.
A good match is observed up to 200Hz.
Similar result is obtained for the Y direction.
This therefore indicates that the considered bodies are behaving as solid bodes in the frequency range of interest.
#+begin_src matlab
%% Load frequency response matrix
load('frf_matrix.mat', 'freqs', 'frf');
%% Load Accelerometer positions
acc_pos = readtable('mat/acc_pos.txt', 'ReadVariableNames', false);
acc_pos = table2array(acc_pos(:, 1:4));
[~, i] = sort(acc_pos(:, 1));
acc_pos = acc_pos(i, 2:4);
#+end_src
#+begin_src matlab
@ -798,9 +754,9 @@ exportFig('figs/modal_solid_body_comp_x_dir.pdf', 'width', 'full', 'height', 'ta
** 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).
The goal here is to link these $4 \times 3 = 12$ measurements to the 6 DOFs of the solid body expressed in the frame $\{O\}$.
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 $\{O\}$.
#+begin_src latex :file modal_local_to_global_coordinates.pdf
\newcommand\irregularcircle[2]{% radius, irregularity
@ -841,27 +797,9 @@ The goal here is to link these $4 \times 3 = 12$ measurements to the 6 DOFs of t
#+RESULTS:
[[file:figs/modal_local_to_global_coordinates.png]]
Let's consider the motion of the rigid body defined by its displacement $\delta \vec{p}$ and rotation $\delta\vec{\Omega}$ with respect to the reference frame $\{O\}$.
Writing Eq. eqref:eq:modal_p1_p2_motion for the four displacement sensors in a matrix form gives eqref:eq:modal_cart_to_acc.
From the figure ref:fig:modal_local_to_global_coordinates, we can write:
\begin{align*}
\delta \vec{p}_1 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_1\\
\delta \vec{p}_2 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_2\\
\delta \vec{p}_3 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_3\\
\delta \vec{p}_4 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_4
\end{align*}
With
\begin{equation}
\bm{\delta\Omega} = \begin{bmatrix}
0 & -\delta\Omega_z & \delta\Omega_y \\
\delta\Omega_z & 0 & -\delta\Omega_x \\
-\delta\Omega_y & \delta\Omega_x & 0
\end{bmatrix}
\end{equation}
We can rearrange the equations in a matrix form:
\begin{equation}
\begin{equation}\label{eq:modal_cart_to_acc}
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
@ -877,7 +815,11 @@ We can rearrange the equations in a matrix form:
\end{array}\right]
\end{equation}
and then we obtain the velocity and rotation of the solid in the wanted 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.
\begin{equation}
\left[\begin{array}{c}
\delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
@ -895,41 +837,9 @@ and then we obtain the velocity and rotation of the solid in the wanted frame $\
\end{array}\right] \label{eq:modal_determine_global_disp}
\end{equation}
#+begin_important
Using equation eqref:eq:modal_determine_global_disp, we can determine the motion of the solid body expressed in a chosen frame $\{O\}$ using the accelerometers attached to it.
The inversion is equivalent to resolving a mean square problem.
#+end_important
** Frequency Response Matrix expressed at the Center of Mass
<<ssec:modal_frf_com>>
**** What reference frame to choose?
The question we wish here to answer is how to choose the reference frame $\{O\}$ in which the DOFs of the solid bodies are defined.
The possibles choices are:
- *One frame for each solid body* which is located at its center of mass
- *One common frame*, for instance located at the point of interest ($270mm$ above the Hexapod)
- *Base located at the joint position*: this is where we want to see the motion and estimate stiffness
#+name: tab:modal_frame_comparison
#+caption: Advantages and disadvantages for the choice of reference frame
#+attr_latex: :environment tabularx :width \linewidth :align XXX
#+attr_latex: :center t :booktabs t :font \scriptsize
| Chosen Frame | Advantages | Disadvantages |
|--------------------------+-----------------------------------------------------+------------------------------------------------------|
| Frames at CoM | Physically, it makes more sense | How to compare the motion of the solid bodies? |
| Common Frame | We can compare the motion of each solid body | Small $\theta_{x, y}$ may result in large $T_{x, y}$ |
| Frames at joint position | Directly gives which joint direction can be blocked | How to choose the joint position? |
The choice of the frame depends of what we want to do with the data.
One of the goals is to compare the motion of each solid body to see which relative DOFs between solid bodies can be neglected, that is to say, which joint between solid bodies can be regarded as perfect (and this in all the frequency range of interest).
Ideally, we would like to have the same number of degrees of freedom than the number of identified modes.
In the next sections, we will express the FRF matrix in the different frames.
**** Center of Mass of each solid body
From solidworks, we can export the position of the center of mass of each solid body considered.
These are summarized in Table ref:tab:modal_com_solid_bodies
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.
#+begin_src matlab
%% Extract the CoM of considered solid bodies
@ -941,7 +851,7 @@ data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation st
#+end_src
#+name: tab:modal_com_solid_bodies
#+caption: Center of mass of considered solid bodies
#+caption: Center of mass of considered solid bodies with respect to the "point of interest"
#+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX
#+attr_latex: :center t :booktabs t
#+RESULTS:
@ -954,17 +864,10 @@ data2orgtable(1000*model_com', {'Bottom Granite', 'Top granite', 'Translation st
| Spindle | 0 | 0 | -580 |
| Hexapod | -4 | 6 | -319 |
Using eqref:eq:modal_determine_global_disp, the frequency response matrix $\bm{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) at the center of mass of each solid body can be computed from the initial FRF matrix $\bm{H}$.
**** Computation
First, we initialize a new FRF matrix which is an $n \times p \times q$ with:
- $n$ is the number of DOFs of the considered 6 solid-bodies: $6 \times 6 = 36$
- $p$ is the number of excitation inputs: $3$
- $q$ is the number of frequency points $\omega_i$
#+begin_important
For each frequency point $\omega_i$, the FRF matrix is a $n\times p$ matrix:
\begin{equation}
\text{FRF}_\text{CoM}(\omega_i) = \begin{bmatrix}
\begin{equation}\label{eq:modal_frf_matrix_com}
\bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
@ -976,10 +879,6 @@ For each frequency point $\omega_i$, the FRF matrix is a $n\times p$ matrix:
\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
\end{bmatrix}
\end{equation}
where 1, 2, ..., 6 corresponds to the 6 solid bodies.
#+end_important
Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in their center of mass.
#+begin_src matlab
%% Frequency Response Matrix - Response expressed at the CoM of the solid bodies
@ -1018,29 +917,13 @@ save('mat/frf_com.mat', 'FRFs_CoM');
save('matlab/mat/frf_com.mat', 'FRFs_CoM');
#+end_src
** Verify that we find the original FRF from the FRF in the global coordinates
** Verification of solid body assumption
<<ssec:modal_solid_body_assumption>>
We have computed the Frequency Response Functions Matrix representing the response of the 6 solid bodies in their 6 DOFs with respect to their center of mass.
From the response of one solid body along its 6 DoFs (from $\bm{H}_{\text{CoM}}$), and using eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any location, in particular at the location of the accelerometers fixed to this solid body.
From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement, supposing that this stage is a 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.
We can then compare the result with the original measurements.
This will help us to determine if:
- the previous inversion used is correct
- the solid body assumption is correct in the frequency band of interest
From the translation $\delta p$ and rotation $\delta \Omega$ of a solid body and the positions $p_i$ of the accelerometers attached to it, we can compute the response that would have been measured by the accelerometers using the following formula:
\begin{align*}
\delta p_1 &= \delta p + \delta\Omega p_1\\
\delta p_2 &= \delta p + \delta\Omega p_2\\
\delta p_3 &= \delta p + \delta\Omega p_3\\
\delta p_4 &= \delta p + \delta\Omega p_4
\end{align*}
Thus, we can obtain the FRF matrix =FRFs_A= that gives the responses of the accelerometers to the forces applied by the hammer.
It is implemented in matlab as follow:
#+begin_src matlab
FRFs_A = zeros(size(frf));
@ -1064,13 +947,10 @@ for exc_dir = 1:3
end
#+end_src
We then compare the original FRF measured for each accelerometer =FRFs= with the "recovered" FRF =FRFs_A= from the global FRF matrix in the common frame.
The FRF for the 4 accelerometers on the Hexapod are compared in Figure ref:fig:modal_comp_acc_solid_body_frf.
All the FRF are matching very well in all the frequency range displayed.
# The FRF for accelerometers located on the translation stage are compared on figure ref:fig:modal_recovered_frf_comparison_ty.
# The FRF are matching well until 100Hz.
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).
#+begin_src matlab :exports none :results none
%% Comparaison of the original accelerometer response and reconstructed response from the solid body response
@ -1083,7 +963,7 @@ exc_dir = 1; % Excited direction
accs_i = solids.(solid_names{solid_i}); % Accelerometers fixed to this solid body
figure;
tiledlayout(2, 2, 'TileSpacing', 'Compact', 'Padding', 'None');
tiledlayout(2, 2, 'TileSpacing', 'Tight', 'Padding', 'None');
for i = 1:length(accs_i)
acc_i = accs_i(i);
@ -1111,9 +991,10 @@ for i = 1:length(accs_i)
set(gca, 'YTickLabel',[]);
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlim([5, 200]); ylim([1e-6, 1e-1]);
legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'log');
xlim([0, 200]); ylim([1e-6, 3e-2]);
xticks([0:20:200]);
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 1);
end
#+end_src
@ -1122,17 +1003,10 @@ 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 and reconstructed response from the solid body response. For accelerometers 1 to 4 corresponding to the hexapod.
#+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.
#+RESULTS:
[[file:figs/modal_comp_acc_solid_body_frf.png]]
#+begin_important
The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well.
This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest.
This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
#+end_important
* Modal Analysis
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/modal_3_analysis.m

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@ -1,4 +1,4 @@
% Created 2024-06-26 Wed 16:06
% Created 2024-06-26 Wed 17:58
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -63,7 +63,7 @@ Obtained force and acceleration signals are shown in Section \ref{ssec:modal_mea
Three equipment are key to perform a good modal analysis.
First, \emph{accelerometers} are used to measure the response of the structure.
Here, 3-axis accelerometers\footnote{PCB 393B05. Sensitivity is \(10\,V/g\), measurement range is \(0.5\,g\) and bandwidth is \(0.7\) to \(450\,\text{Hz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
These accelerometers are glued to the micro-station using a thin layer of wax for best results (see \cite[chapt. 3.5.7]{ewins00_modal}).
These accelerometers are glued to the micro-station using a thin layer of wax for best results \cite[chapt. 3.5.7]{ewins00_modal}.
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
@ -88,8 +88,8 @@ These accelerometers are glued to the micro-station using a thin layer of wax fo
\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 hard metallic one to soft plastic).
The softer tip has been found to give best results as it permits to inject more energy in the low frequency range where the coherence was low, such that the overall coherence was improved.
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 \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.
@ -97,28 +97,28 @@ Finally, an \emph{acquisition system}\footnote{OROS OR36. 24bits signal-delta AD
\label{ssec:modal_test_preparation}
In order to obtain meaningful results, the modal analysis of the micro-station in performed \emph{in-situ}.
To do so, the local position control of all the micro-station stages 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 \(T_y\) stage.
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 and the sample platform have been removed in order to reduce the complexity of the dynamics and also because the active stabilization stage and the sample will be added in the multi-body model afterwards.
The top part representing the active stabilization stage has been disassembled 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, with \(n\) 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 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 \emph{point measurement} (\(H_{jj}\)), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix \(\mathbf{H}\) \cite[chapt. 5.2]{ewins00_modal}.
Therefore, a minimum set of \(n\) frequency response functions needs to be measured.
This can be done either by measuring the response \(X_{j}\) at a fixed DoF \(j\) while applying forces \(F_{i}\) for at all \(n\) considered DoF, or by applying a force \(F_{k}\) at a fixed DoF and measuring the response \(X_{i}\) for all \(n\) DoF.
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 advice 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 excited.
It is however not advised to measure only one row or one column as one or more modes may be missed by an unfortunate choice of force or acceleration measured locations.
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.
\section{Location of the Accelerometers}
\label{ssec:modal_accelerometers}
The location of the accelerometers fixed to the micro-station is essential as it defines at which points the dynamics is identified.
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}.
@ -172,7 +172,7 @@ Hexapod & 64 & -64 & -270\\
\begin{figure}
\begin{figure}[htbp]
\begin{subfigure}{0.49\textwidth}
\begin{center}
\includegraphics[scale=1,height=6cm]{figs/modal_accelerometers_ty.jpg}
@ -191,13 +191,13 @@ Hexapod & 64 & -64 & -270\\
\section{Hammer Impacts}
\label{ssec:modal_hammer_impacts}
The impact location corresponds to the location of accelerometer \(11\) fixed to the translation stage.
The chosen location of the hammer impact corresponds to the location of accelerometer number \(11\) fixed to the translation stage.
It was chosen to match the location of one accelerometer, because a \emph{point measurement} (i.e. a measurement of \(H_{kk}\)) is necessary to be able to reconstruct the full FRF matrix \cite{ewins00_modal}.
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 were chosen in order to properly excite all the modes in the frequency band of interest and because it provides good coherence for all the accelerometers.
This excitation point and the three considered directions allows to properly excite all the modes in the frequency band of interest and to provide good coherence for all the accelerometers as will be shown in the next section.
This excitation point corresponds to the location of one accelerometer, because a \emph{point measurement} is necessary to be able to reconstruct the full FRF matrix \cite{ewins00_modal}.
\begin{figure}
\begin{figure}[htbp]
\begin{subfigure}{0.33\textwidth}
\begin{center}
\includegraphics[scale=1,width=0.8\linewidth]{figs/modal_impact_x.jpg}
@ -245,7 +245,7 @@ Conclusions based on the time domain signals can be clearly seen in the frequenc
\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})}
\end{figure}
The frequency response function 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 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}.
Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which corresponds to the frequency range of interest.
@ -267,17 +267,17 @@ Good coherence is obtained from \(20\,\text{Hz}\) to \(200\,\text{Hz}\) which co
\chapter{Frequency Analysis}
\label{sec:modal_frf_processing}
The measurements have been conducted and a \(n \times p \times q\) Frequency Response Functions Matrix has been computed with:
All measurements where conducted and a \(n \times p \times q\) Frequency Response Functions Matrix were computed with:
\begin{itemize}
\item \(n\): the number of output measured accelerations: \(23 \times 3 = 69\) (23 accelerometers measuring 3 directions each)
\item \(p\): the number of input force excitations: \(3\)
\item \(q\): the number of frequency points \(\omega_i\)
\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 \(\omega_{i}\)
\end{itemize}
Thus, the FRF matrix is an \(69 \times 3 \times 801\) matrix.
For each frequency point \(\omega_i\), a 2D matrix is obtained that links the 3 force inputs to the 69 output accelerations:
\begin{equation}
\text{FRF}(\omega_i) = \begin{bmatrix}
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}.
\begin{equation}\label{eq:modal_frf_matrix_raw}
\mathbf{H}(\omega_i) = \begin{bmatrix}
\frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\
\frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\
\frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\
@ -292,21 +292,17 @@ 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 measures stages have to behave as rigid bodies in the frequency band of interest.
This
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}).
\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 go from the accelerometers DoFs to the solid body 6 DoFs (three translations and three rotations), some computations have to be performed.
This is explained in Section \ref{ssec:modal_acc_to_solid_dof}.
Finally, the \(69 \times 3 \times 801\) frequency response matrix is reduced to a \(36 \times 3 \times 801\) frequency response matrix where the motion of each solid body is expressed with respect to the CoM of the solid body (Section \ref{ssec:modal_frf_com}).
To validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}.
To further validate this reduction of DoF and the solid body assumption, the frequency response function at the accelerometer location are synthesized from the reduced frequency response matrix and are compared with the initial measurements in Section \ref{ssec:modal_solid_body_assumption}.
\section{First verification of the solid body assumption}
\label{ssec:modal_solid_body_first_check}
In this section, it is shown that two accelerometers fixed to a rigid body at positions \(\vec{p}_1\) and \(\vec{p}_2\) in such a way that \(\vec{p}_2 = \vec{p}_1 + \alpha \vec{x}\) will measure the same value in the \(\vec{x}\) direction.
Such situation is illustrated on Figure \ref{fig:modal_aligned_accelerometers} and is the case for many accelerometers as shown in Figure \ref{fig:modal_location_accelerometers}.
In this section, it is shown that two accelerometers fixed to a \emph{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}.
\begin{figure}[htbp]
\centering
@ -314,35 +310,49 @@ Such situation is illustrated on Figure \ref{fig:modal_aligned_accelerometers} a
\caption{\label{fig:modal_aligned_accelerometers}Aligned measurement of the motion of a solid body}
\end{figure}
The motion of the rigid body of figure \ref{fig:modal_aligned_accelerometers} can be defined by its displacement \(\delta \vec{p}\) and rotation \(\vec{\Omega}\) with respect to a reference frame \(\{O\}\).
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 motions at points \(1\) and \(2\) are:
\begin{align*}
\delta \vec{p}_1 &= \delta \vec{p} + \vec{\Omega} \times \vec{p}_1 \\
\delta \vec{p}_2 &= \delta \vec{p} + \vec{\Omega} \times \vec{p}_2
\end{align*}
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}.
Taking only the \(x\) direction:
\begin{align*}
\delta p_{x1} &= \delta p_x + \Omega_y p_{z1} - \Omega_z p_{y1} \\
\delta p_{x2} &= \delta p_x + \Omega_y p_{z2} - \Omega_z p_{y2}
\end{align*}
\begin{subequations}\label{eq:modal_p1_p2_motion}
\begin{align}
\vec{\delta} p_{1} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{1} \\
\vec{\delta} p_{2} &= \vec{\delta} p + \bm{\delta \Omega} \cdot \vec{p}_{2}
\end{align}
\end{subequations}
However, we have \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\) because of the co-linearity of the two sensors in the \(x\) direction, and thus we obtain:
\begin{equation}
\begin{equation}\label{eq:modal_rotation_matrix}
\bm{\delta\Omega} = \begin{bmatrix}
0 & -\delta\Omega_z & \delta\Omega_y \\
\delta\Omega_z & 0 & -\delta\Omega_x \\
-\delta\Omega_y & \delta\Omega_x & 0
\end{bmatrix}
\end{equation}
Considering only the \(x\) direction, equation \eqref{eq:modal_p1_p2_x_motion} is obtained.
\begin{subequations}\label{eq:modal_p1_p2_x_motion}
\begin{align}
\delta p_{x1} &= \delta p_x + \delta \Omega_y p_{z1} - \delta \Omega_z p_{y1} \\
\delta p_{x2} &= \delta p_x + \delta \Omega_y p_{z2} - \delta \Omega_z p_{y2}
\end{align}
\end{subequations}
Because the two sensors are co-linearity in the \(x\) direction, \(p_{1y} = p_{2y}\) and \(p_{1z} = p_{2z}\), and \eqref{eq:modal_colinear_sensors_equal} is obtained.
\begin{equation}\label{eq:modal_colinear_sensors_equal}
\boxed{\delta p_{x1} = \delta p_{x2}}
\end{equation}
It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction ``linking'' the two sensors.
It is therefore concluded that two position sensors fixed to a rigid body will measure the same quantity in the direction ``in line'' the two sensors.
Such property can be used to verify that the considered stages are indeed behaving as rigid body in the frequency band of interest.
From Table \ref{tab:modal_position_accelerometers}, we can identify which pair of accelerometers are aligned in the X and Y directions.
From Table \ref{tab:modal_position_accelerometers}, the pairs of accelerometers that aligned in the X and Y directions can be identified.
The response in the X direction of pairs of sensors aligned in the X direction are compared in Figure \ref{fig:modal_solid_body_comp_x_dir}.
Good match is observed up to 200Hz.
A good match is observed up to 200Hz.
Similar result is obtained for the Y direction.
This therefore indicates that the considered bodies are behaving as solid bodes in the frequency range of interest.
\begin{figure}[htbp]
\centering
@ -352,9 +362,9 @@ Similar result is obtained for the Y direction.
\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).
The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of the solid body expressed in the frame \(\{O\}\).
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 \(\{O\}\).
\begin{figure}[htbp]
\centering
@ -362,27 +372,9 @@ The goal here is to link these \(4 \times 3 = 12\) measurements to the 6 DOFs of
\caption{\label{fig:modal_local_to_global_coordinates}Schematic of the measured motions of a solid body}
\end{figure}
Let's consider the motion of the rigid body defined by its displacement \(\delta \vec{p}\) and rotation \(\delta\vec{\Omega}\) with respect to the reference frame \(\{O\}\).
Writing Eq. \eqref{eq:modal_p1_p2_motion} for the four displacement sensors in a matrix form gives \eqref{eq:modal_cart_to_acc}.
From the figure \ref{fig:modal_local_to_global_coordinates}, we can write:
\begin{align*}
\delta \vec{p}_1 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_1\\
\delta \vec{p}_2 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_2\\
\delta \vec{p}_3 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_3\\
\delta \vec{p}_4 &= \delta \vec{p} + \bm{\delta \Omega} \vec{p}_4
\end{align*}
With
\begin{equation}
\bm{\delta\Omega} = \begin{bmatrix}
0 & -\delta\Omega_z & \delta\Omega_y \\
\delta\Omega_z & 0 & -\delta\Omega_x \\
-\delta\Omega_y & \delta\Omega_x & 0
\end{bmatrix}
\end{equation}
We can rearrange the equations in a matrix form:
\begin{equation}
\begin{equation}\label{eq:modal_cart_to_acc}
\left[\begin{array}{ccc|ccc}
1 & 0 & 0 & 0 & p_{1z} & -p_{1y} \\
0 & 1 & 0 & -p_{1z} & 0 & p_{1x} \\
@ -398,7 +390,11 @@ We can rearrange the equations in a matrix form:
\end{array}\right]
\end{equation}
and then we obtain the velocity and rotation of the solid in the wanted 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.
\begin{equation}
\left[\begin{array}{c}
\delta p_x \\ \delta p_y \\ \delta p_z \\ \hline \delta\Omega_x \\ \delta\Omega_y \\ \delta\Omega_z
@ -416,49 +412,9 @@ and then we obtain the velocity and rotation of the solid in the wanted frame \(
\end{array}\right] \label{eq:modal_determine_global_disp}
\end{equation}
\begin{important}
Using equation \eqref{eq:modal_determine_global_disp}, we can determine the motion of the solid body expressed in a chosen frame \(\{O\}\) using the accelerometers attached to it.
The inversion is equivalent to resolving a mean square problem.
\end{important}
\section{Frequency Response Matrix expressed at the Center of Mass}
\label{ssec:modal_frf_com}
\paragraph{What reference frame to choose?}
The question we wish here to answer is how to choose the reference frame \(\{O\}\) in which the DOFs of the solid bodies are defined.
The possibles choices are:
\begin{itemize}
\item \textbf{One frame for each solid body} which is located at its center of mass
\item \textbf{One common frame}, for instance located at the point of interest (\(270mm\) above the Hexapod)
\item \textbf{Base located at the joint position}: this is where we want to see the motion and estimate stiffness
\end{itemize}
\begin{table}[htbp]
\centering
\scriptsize
\begin{tabularx}{\linewidth}{XXX}
\toprule
Chosen Frame & Advantages & Disadvantages\\
\midrule
Frames at CoM & Physically, it makes more sense & How to compare the motion of the solid bodies?\\
Common Frame & We can compare the motion of each solid body & Small \(\theta_{x, y}\) may result in large \(T_{x, y}\)\\
Frames at joint position & Directly gives which joint direction can be blocked & How to choose the joint position?\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_frame_comparison}Advantages and disadvantages for the choice of reference frame}
\end{table}
The choice of the frame depends of what we want to do with the data.
One of the goals is to compare the motion of each solid body to see which relative DOFs between solid bodies can be neglected, that is to say, which joint between solid bodies can be regarded as perfect (and this in all the frequency range of interest).
Ideally, we would like to have the same number of degrees of freedom than the number of identified modes.
In the next sections, we will express the FRF matrix in the different frames.
\paragraph{Center of Mass of each solid body}
From solidworks, we can export the position of the center of mass of each solid body considered.
These are summarized in Table \ref{tab:modal_com_solid_bodies}
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.
\begin{table}[htbp]
\centering
@ -474,23 +430,14 @@ Spindle & 0 & 0 & -580\\
Hexapod & -4 & 6 & -319\\
\bottomrule
\end{tabularx}
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies}
\caption{\label{tab:modal_com_solid_bodies}Center of mass of considered solid bodies with respect to the ``point of interest''}
\end{table}
Using \eqref{eq:modal_determine_global_disp}, the frequency response matrix \(\bm{H}_\text{CoM}\) \eqref{eq:modal_frf_matrix_com} expressing the response \(D_i\) (\(i\) from \(1\) to \(6\) for the \(6\) considered solid bodies) at the center of mass of each solid body can be computed from the initial FRF matrix \(\bm{H}\).
\paragraph{Computation}
First, we initialize a new FRF matrix which is an \(n \times p \times q\) with:
\begin{itemize}
\item \(n\) is the number of DOFs of the considered 6 solid-bodies: \(6 \times 6 = 36\)
\item \(p\) is the number of excitation inputs: \(3\)
\item \(q\) is the number of frequency points \(\omega_i\)
\end{itemize}
\begin{important}
For each frequency point \(\omega_i\), the FRF matrix is a \(n\times p\) matrix:
\begin{equation}
\text{FRF}_\text{CoM}(\omega_i) = \begin{bmatrix}
\begin{equation}\label{eq:modal_frf_matrix_com}
\bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
@ -502,54 +449,25 @@ For each frequency point \(\omega_i\), the FRF matrix is a \(n\times p\) matrix:
\frac{D_{6,R_z}}{F_x}(\omega_i) & \frac{D_{6,R_z}}{F_y}(\omega_i) & \frac{D_{6,R_z}}{F_z}(\omega_i)
\end{bmatrix}
\end{equation}
where 1, 2, \ldots{}, 6 corresponds to the 6 solid bodies.
\end{important}
Then, as we know the positions of the accelerometers on each solid body, and we have the response of those accelerometers, we can use the equations derived in the previous section to determine the response of each solid body expressed in their center of mass.
\section{Verify that we find the original FRF from the FRF in the global coordinates}
\section{Verification of solid body assumption}
\label{ssec:modal_solid_body_assumption}
We have computed the Frequency Response Functions Matrix representing the response of the 6 solid bodies in their 6 DOFs with respect to their center of mass.
From the response of one solid body along its 6 DoFs (from \(\bm{H}_{\text{CoM}}\)), and using \eqref{eq:modal_cart_to_acc}, it is possible to compute the response of the same solid body at any location, in particular at the location of the accelerometers fixed to this solid body.
From the response of one body in its 6 DOFs, we should be able to compute the FRF of each of its accelerometer fixed to it during the measurement, supposing that this stage is a 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.
We can then compare the result with the original measurements.
This will help us to determine if:
\begin{itemize}
\item the previous inversion used is correct
\item the solid body assumption is correct in the frequency band of interest
\end{itemize}
From the translation \(\delta p\) and rotation \(\delta \Omega\) of a solid body and the positions \(p_i\) of the accelerometers attached to it, we can compute the response that would have been measured by the accelerometers using the following formula:
\begin{align*}
\delta p_1 &= \delta p + \delta\Omega p_1\\
\delta p_2 &= \delta p + \delta\Omega p_2\\
\delta p_3 &= \delta p + \delta\Omega p_3\\
\delta p_4 &= \delta p + \delta\Omega p_4
\end{align*}
Thus, we can obtain the FRF matrix \texttt{FRFs\_A} that gives the responses of the accelerometers to the forces applied by the hammer.
It is implemented in matlab as follow:
We then compare the original FRF measured for each accelerometer \texttt{FRFs} with the ``recovered'' FRF \texttt{FRFs\_A} from the global FRF matrix in the common frame.
The FRF for the 4 accelerometers on the Hexapod are compared in Figure \ref{fig:modal_comp_acc_solid_body_frf}.
All the FRF are matching very well in all the frequency range displayed.
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).
\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 and reconstructed response from the solid body response. For accelerometers 1 to 4 corresponding to the hexapod.}
\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.}
\end{figure}
\begin{important}
The reduction of the number of degrees of freedom from 69 (23 accelerometers with each 3DOF) to 36 (6 solid bodies with 6 DOF) seems to work well.
This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest.
This valid the fact that a multi-body model can be used to represent the dynamics of the micro-station.
\end{important}
\chapter{Modal Analysis}
\label{sec:modal_analysis}
The goal here is to extract the modal parameters describing the modes of station being studied, namely: