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Thomas Dehaeze 2025-02-04 15:38:20 +01:00
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@ -226,7 +226,7 @@ The obtained force and acceleration signals are described in Section ref:ssec:mo
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.
Here, 3-axis accelerometers[fn:modal_1] shown in figure ref:fig:modal_accelero_M393B05 are used.
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
@ -253,11 +253,11 @@ These accelerometers were glued to the micro-station using a thin layer of wax f
#+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 manner.
Then, an /instrumented hammer/[fn:modal_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 response accelerations in a synchronized manner and with sufficiently low noise.
Finally, an /acquisition system/[fn:modal_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>>
@ -678,7 +678,7 @@ Writing this in matrix form for the four points gives eqref:eq:modal_cart_to_acc
\end{array}\right]
\end{equation}
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].
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:modal_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 better approximation of the motion of a solid body.
@ -1079,7 +1079,7 @@ The levelers were then better adjusted.
#+attr_latex: :width 0.6\linewidth
[[file:figs/modal_airlock_picture.jpg]]
The modal parameter extraction is made using a proprietary software[fn:4].
The modal parameter extraction is made using a proprietary software[fn:modal_4].
For each mode $r$ (from $1$ to the number of considered modes $m=16$), it outputs the frequency $\omega_r$, the damping ratio $\xi_r$, the eigenvectors $\{\phi_{r}\}$ (vector of complex numbers with a size equal to the number of measured acrshort:dof $n=69$, see equation eqref:eq:modal_eigenvector) and a scaling factor $a_r$.
\begin{equation}\label{eq:modal_eigenvector}
@ -1330,8 +1330,8 @@ colors = colororder;
* Footnotes
[fn:5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.
[fn:4]NVGate software from OROS company.
[fn:3]OROS OR36. 24bits signal-delta ADC.
[fn:2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$
[fn:1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.
[fn:modal_5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead.
[fn:modal_4]NVGate software from OROS company.
[fn:modal_3]OROS OR36. 24bits signal-delta ADC.
[fn:modal_2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$
[fn:modal_1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.