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.
The approach of tuning the multi-body model from 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.
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.
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.
Here, 3-axis accelerometers\footnote{PCB 356B18. Sensitivity is \(1\,V/g\), measurement range is \(\pm5\,g\) and bandwidth is \(0.5\) to \(5\,\text{kHz}\).} shown in figure \ref{fig:modal_accelero_M393B05} are used.
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.
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.
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 as 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 \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 \(n^2\) 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 \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 energized.
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 \acrshort{dof} can be considered per 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}).
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}.
This excitation point with the three considered directions allows to properly energize 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.
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).
\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})}
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{\label{fig:modal_frf_coh_acc_force}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})}
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 \(6\times6=36\) degrees of freedom are of interest.
In order to be able to perform this reduction of measured \acrshort{dof}, the rigid body assumption first needs to be verified (Section \ref{ssec:modal_solid_body_first_check}).
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 \(69\times3\times801\) frequency response matrix is then reduced to a \(36\times3\times801\) 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 \acrshort{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 \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}.
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{\label{fig:modal_solid_body_comp_x_dir}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.}
\end{figure}
\section{From accelerometer DOFs to solid body DOFs}
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).
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 \acrshort{frf} matrix \(\bm{H}\).
From the response of one solid body along its 6 \acrshort{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 3acrshort:dof) to 36 (6 solid bodies with 6 \acrshort{dof}).
\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.}
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 \textbf{curve-fitting a theoretical expression for an individual \acrshort{frf} to the actual measured data}.
In order to validate the quality of the modal model, we will synthesize the \acrshort{frf} matrix from the modal model and compare it with the \acrshort{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.
\section{Determine the number of modes}
\label{ssec:modal_number_of_modes}
\paragraph{Singular Value Decomposition - Modal Indication Function}
The Mode Indicator Functions are usually used on \(n\times p\)\acrshort{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 \acrshort{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 \textbf{determine the number of modes} present in a given frequency range, to \textbf{identify repeated natural frequencies} and to pre-process the \acrshort{frf} data prior to modal analysis.
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.
We choose to use the ``broad band'' method as it permits to identify the modal parameters using all the \acrshort{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 \acrshort{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 \textbf{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.
It seems that the modal analysis software makes the \textbf{assumption} of viscous damping for the model with which it tries to fit the \acrshort{frf} measurements.
\caption{\label{fig:modal_mode_animations}Instrumentation used for the modal analysis}
\end{figure}
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}).
\item\(A_{pqr}=\frac{\psi_{pr}\psi_{qr}}{M_{A_r}}\), \(M_{A_r}\) is called ``Modal A''
\item\(\psi_{pr}\): scaled modal coefficient for output DOF \(p\), mode \(r\)
\item\(\lambda_r\): complex modal frequency
\end{itemize}
From the modal software documentation:
\begin{quote}
\textbf{Modal A}
Scaling constant for a complex mode. It has the same properties as modal mass for normal modes (undamped or proportionally damped cases). Assuming
\begin{itemize}
\item\(\psi_{pr}\) = Modal coefficient for measured degree of freedom p and mode r
\item\(\psi_{qr}\) = Modal coefficient for measured degree of freedom q and mode r
\item\(A_{pqr}\) = Residue for measured degree of freedom p, measured degree of q and mode r
\item\(M_{Ar}\) = Modal A of mode r
\end{itemize}
Then
\[ A_{pqr}=\frac{\psi_{pr}\psi_{qr}}{M_{Ar}}\]
\textbf{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
\begin{itemize}
\item\(M_{Ar}\) = Modal A of mode r
\item\(\lambda_r\) = System pole of mode r
\end{itemize}
Then
\[ M_{Br}=-\lambda_r M_{Ar}\]
\end{quote}
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}.
