Add some comments on modal analysis

index.org

@@ -371,18 +371,32 @@ To estimate the PSD of the position error $\epsilon$ and thus the RMS residual m
 <<sec:micro_station_dynamics>>
 
 ** Introduction                                                      :ignore:
-As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be coupled with the dynamics of the nano-hexapod and thus is very important for both the design of the nano-hexapod and the controller.
+As explained before, it is very important to have a good estimation of the micro-station dynamics as it will be coupled with the dynamics of the nano-hexapod and thus is very important for both the design of the nano-hexapod and controller.
+
 The estimated dynamics will also be used to tune the developed multi-body model of the micro-station with which the simulations will be performed.
 
 All the measurements performed on the micro-station are detailed in [[https://tdehaeze.github.io/meas-analysis/][this]] document and summarized in the following sections.
 
 
+The general procedure to identify the dynamics of the micro-station is shown in Figure [[fig:vibration_analysis_procedure]].
+
+The steps are:
+1. extract a Response Model (Frequency Response Functions) from measurements
+2. convert the Response Model into a Modal Model (Natural Frequencies and Mode Shapes)
+3. extract a Spatial Model from the Modal Model (Mass/Damping/Stiffness matrices)
+
+#+name: fig:vibration_analysis_procedure
+#+caption: Vibration Analysis Procedure
+[[file:figs/vibration_analysis_procedure.png]]
+
+The extraction of the Spatial Model (3rd step) was not performed as it requires a lot of time and was not judge necessary.
+
 ** Setup
 <<sec:id_setup>>
 
-To measure the dynamics of such complicated system, it as been chosen to perform a full modal analysis.
+To measure the dynamics of such complicated system, it as been chosen to perform a modal analysis.
 
-To limit the number of degrees of freedom to be measured, we suppose that in the frequency range of interest (DC-300Hz), each of the positioning stage is behaving as a solid body.
+To limit the number of degrees of freedom to be measured, we suppose that in the frequency range of interest (DC-300Hz), each of the positioning stage is behaving as a *solid body*.
 Thus, to fully describe the dynamics of the station, we (only) need to measure 6 degrees of freedom on each of the positioning stage (that is 36 degrees of freedom for the 6 solid bodies).
 
 
@@ -402,9 +416,9 @@ The measurement thus consists of:
   - 3 on top of the spindle
   - 4 on top of the hexapod
 
-In total, 69 degrees of freedom are measured (23 tri axis accelerometers) which is way more that what was required.
+In total, 69 degrees of freedom are measured (23 tri axis accelerometers) which is more that what was required.
 
-We chose to have some redundancy in the measurement to be able to verify that the solid-body assumption was correct for each of the stage.
+We chose to have some redundancy in the measurement to be able to verify that the solid-body assumption is correct for each of the stage.
 
 #+name: fig:hammer_z
 #+caption: Example of one hammer impact
@@ -422,7 +436,7 @@ From the measurements, we obtain all the transfer functions from forces applied
 Modal shapes and natural frequencies are then computed. Example of mode shapes are shown in Figures [[fig:mode1]] [[fig:mode6]].
 
 #+name: fig:mode1
-#+caption: First mode
+#+caption: First mode that shows a suspension mode, probably due to bad leveling of one Airloc
 [[file:figs/mode1.gif]]
 
 #+name: fig:mode6
@@ -434,8 +448,25 @@ We then reduce the number of degrees of freedom from 69 (23 accelerometers with
 
 From the reduced transfer function matrix, we can re-synthesize the response at the 69 measured degrees of freedom and we find that we have an exact match.
 
-This confirms the fact that the stages are indeed behaving as a *solid body* in the frequency band of interest.
+#+begin_important
+This confirms the fact that the stages are indeed behaving as a solid body in the frequency band of interest.
 This thus means that a multi-body model can be used to represent the dynamics of the micro-station.
+#+end_important
+
+
+Many Frequency Response Functions (FRF) are obtained from the measurements.
+Examples of FRF are shown in Figure [[fig:frf_all_bodies_one_direction]].
+These FRF will be used to compare the dynamics of the multi-body model with the micro-station dynamics.
+
+#+name: fig:frf_all_bodies_one_direction
+#+caption: Frequency Response Function from forces applied by the Hammer in the X direction to the acceleration of each solid body in the X direction
+[[file:figs/frf_all_bodies_one_direction.png]]
+
+** Conclusion
+#+begin_important
+  The modal analysis of the micro-station confirmed the fact that a multi-body model should be able to correctly represents the micro-station dynamics.
+  In Section [[sec:multi_body_model]], the obtained Frequency Response Functions will be used to compare the model dynamics with the micro-station dynamics.
+#+end_important
 
 * Identification of the Disturbances
 <<sec:identification_disturbances>>
@@ -594,6 +625,7 @@ The detector requirement would be:
 - Resolution of $\approx 100nm$ (to be discussed)
 
 ** Conclusion
+#+begin_important
 Main disturbance sources have been identified.
 These disturbances will then be included in the multi-body model.
 
@@ -604,6 +636,7 @@ If heavy/stiff cables are to be fixed to the sample, this should be quantified a
 
 Having better estimation of the disturbances would allows to more precisely estimate the attainable performances.
 This should however not change the conclusion of this study nor significantly change the nano-hexapod design.
+#+end_important
 
 * Multi Body Model
 <<sec:multi_body_model>>