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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. Three type of equipment are essential for a good modal analysis.
First, /accelerometers/ are used to measure the response of the structure. 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]]. 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 #+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_subfigure
#+end_figure #+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). 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. 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 ** Structure Preparation and Test Planing
<<ssec:modal_test_preparation>> <<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{array}\right]
\end{equation} \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. 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. 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. 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 #+attr_latex: :width 0.6\linewidth
[[file:figs/modal_airlock_picture.jpg]] [[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 $a_r$. 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} \begin{equation}\label{eq:modal_eigenvector}
@ -1330,8 +1330,8 @@ colors = colororder;
* Footnotes * Footnotes
[fn:5]As this matrix is in general non-square, the Moore–Penrose inverse can be used instead. [fn:modal_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:modal_4]NVGate software from OROS company.
[fn:3]OROS OR36. 24bits signal-delta ADC. [fn:modal_3]OROS OR36. 24bits signal-delta ADC.
[fn:2]Kistler 9722A2000. Sensitivity of $2.3\,mV/N$ and measurement range of $2\,kN$ [fn:modal_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_1]PCB 356B18. Sensitivity is $1\,V/g$, measurement range is $\pm 5\,g$ and bandwidth is $0.5$ to $5\,\text{kHz}$.