nass-micro-station-measurements/modal-analysis/index.org

Modal Analysis

• Goal
• Type of Model
• Instrumentation Used
• Structure Preparation and Test Planning
  • Structure Preparation
  • Test Planing
  • Location of the Accelerometers
  • Hammer Impacts
• Signal Processing
• Frequency Response Functions and Coherence Results
  • Load Data
  • Raw Force Data
  • Raw Response Data
  • Load Data
  • FRF and Coherence Results
• Mode Shapes
• Obtained Modal Matrices
• Compute the Modal Model
  • Position of the accelerometers
  • Define positions of the accelerometers on matlab
  • Import the modal vectors on matlab
    • Mode1
    • Mode2
    • Mode3
    • Mode4
    • All modes
  • Define a point for each solid body
  • Argand Diagram
  • TEST: animate first mode
  • From 6 translations to translation + rotation
• Problem with AirLoc System
• Spatial Mode Extraction

Goal

The goal is to experimentally extract a Spatial Model (mass, damping, stiffness) of the structure (shown on figure fig:nass_picture) in order to tune the Multi-Body model.

Picture of the ID31 Micro-Station. (1) Granite (2) Translation Stage (3) Tilt Stage (4) Hexapod (5) Dummy Mass

The procedure is represented on figure fig:vibration_analysis_procedure where we go from left to right.

Vibration Analysis Procedure

First, we obtain a Response Model (Frequency Response Functions) from measurements. This is further converted into a Modal Model (Natural Frequencies and Mode Shapes). Finally, this is converted into a Spatial Model with the Mass/Damping/Stiffness matrices.

Theses matrices will be used to tune the Simscape (multi-body) model.

The modes we want to identify are those in the frequency range between 0Hz and 150Hz.

Type of Model

The model that we want to obtain is a multi-body model. It is composed of several solid bodies connected with springs and dampers. The solid bodies are represented with different colors on figure fig:nass_solidworks.

In the simscape model, the solid bodies are:

• the granite (1 or 2 solids)
• the translation stage
• the tilt stage
• the spindle and slip-ring
• the hexapod

CAD view of the ID31 Micro-Station

However, each of the DOF of the system may not be relevant for the modes present in the frequency band of interest. For instance, the translation stage may not vibrate in the Z direction for all the modes identified. Then, we can block this DOF and this simplifies the model.

The modal identification done here will thus permit us to determine which DOF can be neglected.

Instrumentation Used

In order to perform to Modal Analysis and to obtain first a Response Model, the following devices are used:

• An acquisition system (OROS) with 24bits ADCs (figure fig:oros)
• 3 tri-axis Accelerometers (figure fig:accelero_M393B05) with parameters shown on table tab:accelero_M393B05
• An Instrumented Hammer with various Tips (figure fig:instrumented_hammer) (figure fig:hammer_tips)

Acquisition system: OROS

The acquisition system permits to auto-range the inputs (probably using variable gain amplifiers) the obtain the maximum dynamic range. This is done before each measurement. Anti-aliasing filters are also included in the system.

Accelerometer used: M393B05

Sensitivity	10V/g
Measurement Range	0.5 g pk
Broadband Resolution	0.000004 g rms
Frequency Range	0.7 to 450Hz
Resonance Frequency	> 2.5kHz

393B05 Accelerometer Data Sheet

Tests have been conducted to determine the most suitable Hammer tip. This has been found that the softer tip gives the best results. It excites more the low frequency range where the coherence is low, the overall coherence was improved.

Instrumented Hammer

Hammer tips

The accelerometers are glued on the structure.

Structure Preparation and Test Planning

Structure Preparation

All the stages are turned ON. This is done for two reasons:

• Be closer to the real dynamic of the station in used
• If the control system of stages are turned OFF, this would results in very low frequency modes un-identifiable with the current setup, and this will also decouple the dynamics which would not be the case in practice

This is critical for the translation stage and the spindle as their is no stiffness in the free DOF (air-bearing for the spindle for instance).

The alternative would have been to mechanically block the stages with screws, but this may result in changing the modes.

The stages turned ON are:

• Translation Stage
• Tilt Stage
• Spindle and Slip-Ring
• Hexapod

The top part representing the NASS and the sample platform have been removed in order to reduce the complexity of the dynamics and also because this will be further added in the model inside Simscape.

All the stages are moved to their zero position (Ty, Ry, Rz, Slip-Ring, Hexapod).

All other elements have been remove from the granite such as another heavy positioning system.

Test Planing

The goal is to identify the full $N \times N$ FRF matrix (where $N$ is the number of degree of freedom of the system).

However, the principle of reciprocity states that: \[ H_{jk} = \frac{X_j}{F_k} = H_{kj} = \frac{X_k}{F_j} \] Thus, only one column or one line of the matrix has to be identified.

Either we choose to identify $\frac{X_k}{F_i}$ or $\frac{X_i}{F_k}$ for any chosen $k$ and for $i = 1,\ ...,\ N$.

We here choose to identify $\frac{X_i}{F_k}$ for practical reasons:

• it is easier to glue the accelerometers on some stages than to excite this particular stage with the Hammer

The measurement thus consists of:

• always excite the structure at the same location with the Hammer
• Move the accelerometers to measure all the DOF of the structure

Location of the Accelerometers

4 tri-axis accelerometers are used for each solid body.

Only 2 could have been used as only 6DOF have to be measured, however, we have chosen to have some redundancy.

This could also help us identify measurement problems or flexible modes is present.

The position of the accelerometers are:

• 4 on the first granite
• 4 on the second granite (figure fig:accelerometers_granite2_overview)
• 4 on top of the translation stage (figure fig:accelerometers_ty_overview)
• 4 on top of the tilt stage
• 4 on top of the spindle
• 4 on top of the hexapod (figure fig:accelerometers_hexa_overview)

Accelerometers located on the top granite

Accelerometers located on top of the translation stage

Accelerometers located on the Hexapod

Hammer Impacts

Only 3 impact points are used.

The impact points are shown on figures fig:hammer_x, fig:hammer_y and fig:hammer_z.

Hammer Blow in the X direction

Hammer Blow in the Y direction

Hammer Blow in the Z direction

Signal Processing

The measurements are averaged 10 times (figure fig:general_parameters) corresponding to 10 hammer impacts.

General Acquisition Settings

Windowing is used on the force response signals.

A boxcar window (figure fig:window_force) is used for the force signal as once the impact on the structure is done, the measured signal is meaningless.

An exponential window (figure fig:window_response) 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.

Window used for the force signal

Window used for the response signal

Frequency Response Functions and Coherence Results

Load Data

  meas1_raw = load('modal_analysis/raw_data/Measurement1.mat');

Raw Force Data

  <<plt-matlab>>

Raw Force Data from Hammer Blow

  <<plt-matlab>>

Raw Force Data from Hammer Blow - Zoom

Raw Response Data

  <<plt-matlab>>

Raw Acceleration Data from Accelerometer

  <<plt-matlab>>

Raw Acceleration Data from Accelerometer - Zoom

Load Data

  meas1 = load('modal_analysis/frf_coh/Measurement1.mat');

FRF and Coherence Results

  <<plt-matlab>>

Example of one measured FRF

  <<plt-matlab>>

Example of one measured Coherence

Mode Shapes

Multiple modal extraction techniques can be used (SIMO, MIMO, narrow band, wide band, …). First preliminary results on 10 identified modes are presented here.

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Mode 8

Mode 9

Mode 10

Obtained Modal Matrices

From the modal analysis software, we can export the obtained eigen matrices: \[ \begin{bmatrix} \omega_1^2 & & 0 \\ & \ddots & \\ 0 & & \omega_n^2 \end{bmatrix}; \quad \Psi = \begin{bmatrix} & & \\ \{\psi_1\} & \dots & \{\psi_n\} \\ & & \end{bmatrix} \]

where $\bar{\omega}_r^2$ is the $r^\text{th}$ eigenvalue squared and $\{\phi\}_r$ is a description of the corresponding mode shape.

The file containing the eigen frequencies and mode shapes are shown below (for the first mode).

Created by N-Modal
Estimator: cmif
18-Jun-19 16:31:25


Mode 1
freq    = 11.11191Hz
damp    = 10.51401%
modal A =   8.52879e+003-2.29043e+003i
modal B =  -9.64203e+004-6.08978e+005i
Mode matrix of local coordinate [DOF: Re IM]
     1X+:  -9.34637e-002   4.52445e-002
     1Y+:   2.33790e-001   1.41439e-003
     1Z+:  -1.73754e-002   6.02449e-003
     2X+:  -7.42108e-002   3.91543e-002
     2Y+:   2.41566e-001  -1.44869e-003
     2Z+:  -5.99285e-003   2.10370e-003
     4X+:  -1.02163e-001   2.79561e-002
     4Y+:   2.29048e-001   2.89782e-002
     4Z+:  -2.85130e-002   1.77132e-004
     5X+:  -8.77132e-002   3.34081e-002
     5Y+:   2.14182e-001   2.14655e-002
     5Z+:  -1.54521e-002   1.26682e-002
     6X+:  -7.90143e-002   2.42583e-002
     6Y+:   2.20669e-001   2.12738e-002
     6Z+:   4.60755e-002   4.96406e-003
     7X+:  -7.79654e-002   2.58385e-002
     7Y+:   2.06861e-001   3.48019e-002
     7Z+:  -1.78311e-002  -1.29704e-002
     8X+:  -8.49357e-002   3.55200e-002
     8Y+:   2.07470e-001   3.59745e-002
     8Z+:  -7.66974e-002  -3.19813e-003
     9X+:  -7.38565e-002   1.95146e-002
     9Y+:   2.17403e-001   2.01550e-002
     9Z+:  -1.77073e-002  -3.46414e-003
    10X+:  -7.77587e-002   2.36700e-002
    10Y+:   2.35654e-001  -2.14540e-002
    10Z+:   7.94165e-002  -2.45897e-002
    11X+:  -8.17972e-002   2.20583e-002
    11Y+:   2.20906e-001  -4.30164e-003
    11Z+:  -5.60520e-003   3.10187e-003
    12X+:  -8.64261e-002   3.66022e-002
    12Y+:   2.15000e-001  -5.74661e-003
    12Z+:  -1.22622e-001   4.11767e-002
    13X+:  -4.25169e-002   1.56602e-002
    13Y+:   5.31036e-002  -1.73951e-002
    13Z+:  -4.07130e-002   1.26884e-002
    14X+:  -3.85032e-002   1.29431e-002
    14Y+:   5.36716e-002  -1.80868e-002
    14Z+:   1.00367e-001  -3.48798e-002
    15X+:  -4.25524e-002   1.46363e-002
    15Y+:   5.19668e-002  -1.69744e-002
    15Z+:   5.89747e-003  -2.32428e-003
    16X+:  -4.31268e-002   1.38332e-002
    16Y+:   5.07545e-002  -1.53045e-002
    16Z+:  -1.04172e-001   3.17984e-002
    17X+:  -2.69757e-002   9.07955e-003
    17Y+:   3.07837e-002  -9.44663e-003
    17Z+:  -7.63502e-003   1.68203e-003
    18X+:  -3.00097e-002   9.23966e-003
    18Y+:   2.83585e-002  -8.97747e-003
    18Z+:   1.52467e-001  -4.78675e-002
    19X+:  -2.70223e-002   6.16478e-003
    19Y+:   3.06149e-002  -6.25382e-003
    19Z+:  -4.84888e-003   1.93970e-003
    20X+:  -2.90976e-002   7.13184e-003
    20Y+:   3.36738e-002  -7.30875e-003
    20Z+:  -1.66902e-001   3.93419e-002
     3X+:  -9.40720e-002   3.93724e-002
     3Y+:   2.52307e-001   0.00000e+000
     3Z+:  -1.53864e-002  -9.25720e-004
    21X+:  -7.91940e-002   4.39648e-002
    21Y+:   2.04567e-001   9.49987e-003
    21Z+:  -1.56087e-002   7.08838e-003
    22X+:  -1.01070e-001   3.13534e-002
    22Y+:   1.92270e-001   1.80423e-002
    22Z+:   2.93053e-003  -1.97308e-003
    23X+:  -8.86455e-002   4.29906e-002
    23Z+:  -3.38351e-002   1.81362e-003
    23Y-:  -1.90862e-001  -2.53414e-002

Compute the Modal Model

Position of the accelerometers

There are 23 accelerometers:

• 4 on the bottom granite
• 4 on the top granite
• 4 on top of the translation stage
• 4 on the tilt stage
• 3 on top of the spindle
• 4 on top of the hexapod

The coordinates defined in the software are displayed below.

     1  1.0000e-001  1.0000e-001  1.1500e+000      0          Top
     2  1.0000e-001 -1.0000e-001  1.1500e+000      0          Top
     3 -1.0000e-001 -1.0000e-001  1.1500e+000      0          Top
     4 -1.0000e-001  1.0000e-001  1.1500e+000      0          Top
     5  4.0000e-001  4.0000e-001  9.5000e-001      0        inner
     6  4.0000e-001 -4.0000e-001  9.5000e-001      0        inner
     7 -4.0000e-001 -4.0000e-001  9.5000e-001      0        inner
     8 -4.0000e-001  4.0000e-001  9.5000e-001      0        inner
     9  5.0000e-001  5.0000e-001  9.0000e-001      0        outer
    10  5.0000e-001 -5.0000e-001  9.0000e-001      0        outer
    11 -5.0000e-001 -5.0000e-001  9.0000e-001      0        outer
    12 -5.0000e-001  5.0000e-001  9.0000e-001      0        outer
    13  5.5000e-001  5.5000e-001  5.5000e-001      0          top
    14  5.5000e-001 -5.5000e-001  5.5000e-001      0          top
    15 -5.5000e-001 -5.5000e-001  5.5000e-001      0          top
    16 -5.5000e-001  5.5000e-001  5.5000e-001      0          top
    17  9.5000e-001  9.5000e-001  4.0000e-001      0          low
    18  9.5000e-001 -9.5000e-001  4.0000e-001      0          low
    19 -9.5000e-001 -9.5000e-001  4.0000e-001      0          low
    20 -9.5000e-001  9.5000e-001  4.0000e-001      0          low
    21  2.0000e-001  2.0000e-001  8.5000e-001      0          bot
    22  0.0000e+000 -2.0000e-001  8.5000e-001      0          bot
    23 -2.0000e-001  2.0000e-001  8.5000e-001      0          bot

Node number	Solid Body	Location	X	Y	Z
1	Hexapod - Top	-X/-Y	-0.10	-0.10	1.15
2		-X/+Y	-0.10	0.10	1.15
3		+X/+Y	0.10	0.10	1.15
4		+X/-Y	0.10	-0.10	1.15
5	Tilt - Top	-X/-Y	-0.40	-0.40	0.95
6		-X/+Y	-0.40	0.40	0.95
7		+X/+Y	0.40	0.40	0.95
8		+X/-Y	0.40	-0.40	0.95
9	Translation - Top	-X/-Y	-0.50	-0.50	0.90
10		-X/+Y	-0.50	0.50	0.90
11		+X/+Y	0.50	0.50	0.90
12		+X/-Y	0.50	-0.50	0.90
13	Top Granite	-X/-Y	-0.55	-0.50	0.55
14		-X/+Y	-0.55	0.50	0.55
15		+X/+Y	0.55	0.50	0.55
16		+X/-Y	0.55	-0.50	0.55
17	Bottom Granite	-X/-Y	-0.95	-0.90	0.40
18		-X/+Y	-0.95	0.90	0.40
19		+X/+Y	0.95	0.90	0.40
20		+X/-Y	0.95	-0.90	0.40
21	Spindle - Top	-X/-Y	-0.20	-0.20	0.85
22		+0/+Y	0.00	0.20	0.85
23		+X/-Y	0.20	-0.20	0.85

Location of each Accelerometer (using the normal coordinate frame with X aligned with the X ray)

Define positions of the accelerometers on matlab

We define the X-Y-Z position of each sensor. Each line corresponds to one accelerometer, X-Y-Z position in meter.

  positions = [...
    -0.10, -0.10, 1.15 ; ...
    -0.10,  0.10, 1.15 ; ...
     0.10,  0.10, 1.15 ; ...
     0.10, -0.10, 1.15 ; ...
    -0.40, -0.40, 0.95 ; ...
    -0.40,  0.40, 0.95 ; ...
     0.40,  0.40, 0.95 ; ...
     0.40, -0.40, 0.95 ; ...
    -0.50, -0.50, 0.90 ; ...
    -0.50,  0.50, 0.90 ; ...
     0.50,  0.50, 0.90 ; ...
     0.50, -0.50, 0.90 ; ...
    -0.55, -0.50, 0.55 ; ...
    -0.55,  0.50, 0.55 ; ...
     0.55,  0.50, 0.55 ; ...
     0.55, -0.50, 0.55 ; ...
    -0.95, -0.90, 0.40 ; ...
    -0.95,  0.90, 0.40 ; ...
     0.95,  0.90, 0.40 ; ...
     0.95, -0.90, 0.40 ; ...
    -0.20, -0.20, 0.85 ; ...
     0.00,  0.20, 0.85 ; ...
     0.20, -0.20, 0.85 ];

  figure;
  hold on;
  fill3(positions(1:4, 1),   positions(1:4, 2),   positions(1:4, 3),   'k', 'FaceAlpha', 0.5)
  fill3(positions(5:8, 1),   positions(5:8, 2),   positions(5:8, 3),   'k', 'FaceAlpha', 0.5)
  fill3(positions(9:12, 1),  positions(9:12, 2),  positions(9:12, 3),  'k', 'FaceAlpha', 0.5)
  fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
  fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
  fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
  hold off;

Import the modal vectors on matlab

Mode1

  mode1 = [...
    -9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
    -7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
    -9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
    -1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
    -8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
    -7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
    -7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
    -8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
    -7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
    -7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
    -8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
    -8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
    -4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
    -3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
    -4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
    -4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
    -2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
    -3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
    -2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
    -2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
    -7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
    -1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
    -8.86455e-002+j*4.29906e-002, +1.90862e-001+j*2.53414e-002, -3.38351e-002+j*1.81362e-003];

Mode2

  mode2 = [...
    +7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004;
    +8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002;
    +8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002;
    +7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002;
    +6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002;
    +7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001;
    +8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002;
    +6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001;
    +4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004;
    +5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002;
    +5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002;
    +4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002;
    +2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002;
    +2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002;
    +2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003;
    +2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002;
    +1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003;
    +1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001;
    +1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003;
    +1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001;
    +6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002;
    +4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003;
    +6.18293e-002-j*5.99671e-002, +2.37693e-001+j*4.34204e-002, +1.63533e-002+j*6.09161e-002]

Mode3

  mode3 = [...
    +1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002;
    +1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003;
    +1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003;
    +1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003;
    +2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002;
    -7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002;
    -7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003;
    +2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004;
    +2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002;
    -1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003;
    -3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003;
    +2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003;
    +2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003;
    -1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003;
    -1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004;
    +1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003;
    +1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003;
    -1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003;
    -1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004;
    +9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003;
    +3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003;
    +3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003;
    +3.60396e-002-j*2.46238e-002, -1.92038e-002-j*6.65895e-002, +3.57722e-003+j*3.64827e-003];

Mode4

  mode4 = [...
    -1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001;
    -9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002;
    -9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002;
    -9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001;
    +1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003;
    -4.58970e-002-j*1.33904e-002, -9.41660e-002+j*4.99682e-002, +1.28276e-001+j*4.59685e-002;
    -6.01521e-002-j*1.30165e-002, +2.56439e-003-j*6.78141e-002, +5.03428e-002-j*1.59420e-001;
    -1.00895e-002-j*8.80550e-002, +1.26327e-002-j*8.14444e-002, +1.59506e-001-j*2.05360e-001;
    -3.04658e-003-j*1.57921e-001, -8.23501e-002+j*4.82748e-002, +1.69315e-001+j*1.22804e-002;
    -8.25875e-002-j*7.31038e-002, -1.08668e-001+j*3.56364e-002, +8.28567e-002-j*4.49596e-003;
    -1.06792e-001-j*6.95394e-002, +3.77195e-002-j*7.65410e-002, +8.00590e-003-j*2.32461e-002;
    -4.84292e-002-j*1.45790e-001, +1.03862e-002-j*7.31212e-002, +1.78122e-001-j*1.00939e-001;
    -3.49891e-002-j*6.20969e-003, -1.18504e-002-j*1.94225e-002, +4.13007e-002+j*7.67087e-003;
    -3.55795e-002+j*1.16708e-003, -1.68128e-002-j*1.82344e-002, +3.92416e-002-j*3.64434e-002;
    -3.45304e-002+j*3.78185e-003, -7.62559e-003-j*2.24241e-002, +6.28286e-003-j*1.32711e-002;
    -9.95646e-003-j*6.04395e-003, -8.73465e-003-j*2.20807e-002, +3.56946e-002+j*1.69231e-002;
    -9.32661e-003-j*5.51944e-003, -1.91087e-002-j*9.09191e-003, +4.04981e-002+j*8.38685e-004;
    -2.84456e-002+j*4.02762e-003, -2.20044e-002-j*8.86197e-003, +4.43051e-002-j*5.21033e-002;
    -3.27019e-002+j*3.59765e-003, +2.93163e-003-j*2.05064e-002, -1.77289e-002-j*1.29477e-002;
    -1.08474e-002-j*5.78419e-003, +3.86759e-003-j*1.91642e-002, +2.10135e-002+j*3.18051e-002;
    -1.34808e-002-j*9.69121e-003, +1.25218e-002-j*2.71411e-002, +2.76673e-001+j*0.00000e+000;
    +1.96744e-003+j*4.90797e-003, -9.82609e-004-j*3.31065e-002, +1.79246e-001-j*3.33238e-002;
    -1.08728e-002-j*8.80278e-003, -1.15217e-002-j*4.01143e-002, +2.30814e-001-j*8.33151e-002];

All modes

mode_shapes = zeros(23, 3, 10);

mode_shapes(:, :, 1) = [...
-9.34637e-002+j*4.52445e-002, +2.33790e-001+j*1.41439e-003, -1.73754e-002+j*6.02449e-003;
-7.42108e-002+j*3.91543e-002, +2.41566e-001-j*1.44869e-003, -5.99285e-003+j*2.10370e-003;
-9.40720e-002+j*3.93724e-002, +2.52307e-001+j*0.00000e+000, -1.53864e-002-j*9.25720e-004;
-1.02163e-001+j*2.79561e-002, +2.29048e-001+j*2.89782e-002, -2.85130e-002+j*1.77132e-004;
-8.77132e-002+j*3.34081e-002, +2.14182e-001+j*2.14655e-002, -1.54521e-002+j*1.26682e-002;
-7.90143e-002+j*2.42583e-002, +2.20669e-001+j*2.12738e-002, +4.60755e-002+j*4.96406e-003;
-7.79654e-002+j*2.58385e-002, +2.06861e-001+j*3.48019e-002, -1.78311e-002-j*1.29704e-002;
-8.49357e-002+j*3.55200e-002, +2.07470e-001+j*3.59745e-002, -7.66974e-002-j*3.19813e-003;
-7.38565e-002+j*1.95146e-002, +2.17403e-001+j*2.01550e-002, -1.77073e-002-j*3.46414e-003;
-7.77587e-002+j*2.36700e-002, +2.35654e-001-j*2.14540e-002, +7.94165e-002-j*2.45897e-002;
-8.17972e-002+j*2.20583e-002, +2.20906e-001-j*4.30164e-003, -5.60520e-003+j*3.10187e-003;
-8.64261e-002+j*3.66022e-002, +2.15000e-001-j*5.74661e-003, -1.22622e-001+j*4.11767e-002;
-4.25169e-002+j*1.56602e-002, +5.31036e-002-j*1.73951e-002, -4.07130e-002+j*1.26884e-002;
-3.85032e-002+j*1.29431e-002, +5.36716e-002-j*1.80868e-002, +1.00367e-001-j*3.48798e-002;
-4.25524e-002+j*1.46363e-002, +5.19668e-002-j*1.69744e-002, +5.89747e-003-j*2.32428e-003;
-4.31268e-002+j*1.38332e-002, +5.07545e-002-j*1.53045e-002, -1.04172e-001+j*3.17984e-002;
-2.69757e-002+j*9.07955e-003, +3.07837e-002-j*9.44663e-003, -7.63502e-003+j*1.68203e-003;
-3.00097e-002+j*9.23966e-003, +2.83585e-002-j*8.97747e-003, +1.52467e-001-j*4.78675e-002;
-2.70223e-002+j*6.16478e-003, +3.06149e-002-j*6.25382e-003, -4.84888e-003+j*1.93970e-003;
-2.90976e-002+j*7.13184e-003, +3.36738e-002-j*7.30875e-003, -1.66902e-001+j*3.93419e-002;
-7.91940e-002+j*4.39648e-002, +2.04567e-001+j*9.49987e-003, -1.56087e-002+j*7.08838e-003;
-1.01070e-001+j*3.13534e-002, +1.92270e-001+j*1.80423e-002, +2.93053e-003-j*1.97308e-003;
-8.86455e-002+j*4.29906e-002, -3.38351e-002+j*1.81362e-003, +1.90862e-001+j*2.53414e-002];

mode_shapes(:, :, 2) = [...
+7.56931e-002+j*3.61548e-002, +2.07574e-001+j*1.69205e-004, +1.29733e-002-j*6.78426e-004;
+8.58732e-002+j*2.54470e-002, +2.07117e-001-j*1.31755e-003, -2.13788e-003-j*1.24974e-002;
+8.17201e-002+j*2.36079e-002, +2.15927e-001+j*1.61300e-002, -5.48456e-004+j*2.55691e-002;
+7.09825e-002+j*3.66313e-002, +2.09969e-001+j*1.11484e-002, +9.19478e-003+j*3.47272e-002;
+6.23935e-002+j*1.02488e-002, +2.30687e-001-j*3.58416e-003, +3.27122e-002-j*5.85468e-002;
+7.61163e-002-j*2.43630e-002, +2.26743e-001-j*1.15334e-002, -6.20205e-003-j*1.21742e-001;
+8.01824e-002-j*1.94769e-002, +1.97485e-001+j*4.50105e-002, -2.21170e-002+j*9.77052e-002;
+6.19294e-002+j*8.15075e-003, +2.03864e-001+j*4.45835e-002, +2.55133e-002+j*1.36137e-001;
+4.38135e-002+j*7.30537e-002, +2.28426e-001-j*6.58868e-003, +1.16313e-002+j*5.09427e-004;
+5.45770e-002+j*4.34251e-002, +2.50823e-001+j*0.00000e+000, -4.63460e-002-j*4.76868e-002;
+5.50987e-002+j*4.26178e-002, +2.29394e-001+j*5.78236e-002, +1.90158e-002+j*1.09139e-002;
+4.98867e-002+j*7.30190e-002, +2.07871e-001+j*4.57750e-002, +6.69433e-002+j*9.00315e-002;
+2.48819e-002+j*3.03222e-002, -2.56046e-002-j*3.34132e-002, +2.13260e-002+j*2.58544e-002;
+2.45706e-002+j*2.60221e-002, -2.57723e-002-j*3.35612e-002, -5.71282e-002-j*6.61562e-002;
+2.68196e-002+j*2.83888e-002, -2.57263e-002-j*3.29627e-002, -2.11722e-003-j*3.37239e-003;
+2.51442e-002+j*3.32558e-002, -2.54372e-002-j*3.25062e-002, +5.65780e-002+j*7.64142e-002;
+1.62437e-002+j*1.94534e-002, -1.31293e-002-j*2.05924e-002, +1.05274e-003+j*3.59474e-003;
+1.83431e-002+j*2.03836e-002, -1.16818e-002-j*1.86334e-002, -8.66632e-002-j*1.08216e-001;
+1.62553e-002+j*1.79588e-002, -1.28857e-002-j*1.90512e-002, +6.25653e-003+j*4.97733e-003;
+1.63830e-002+j*2.03943e-002, -1.48941e-002-j*2.11717e-002, +8.68045e-002+j*1.16491e-001;
+6.79204e-002-j*5.55513e-002, +2.32871e-001+j*2.33389e-002, +1.34345e-002-j*2.31815e-002;
+4.02414e-002-j*8.38957e-002, +2.35273e-001+j*2.73256e-002, -8.51632e-003-j*7.49635e-003;
+6.18293e-002-j*5.99671e-002, +1.63533e-002+j*6.09161e-002, +2.37693e-001+j*4.34204e-002];

mode_shapes(:, :, 3) = [...
+1.34688e-001-j*6.65071e-002, +1.55316e-002+j*1.01277e-002, -5.88466e-002+j*1.14294e-002;
+1.53934e-001-j*9.76990e-003, +7.17487e-003+j*1.11925e-002, -4.57205e-002+j*7.26573e-003;
+1.61551e-001+j*1.65478e-002, -4.12527e-004-j*5.60909e-002, -9.00640e-003+j*3.50754e-003;
+1.37298e-001-j*5.24661e-002, +1.19427e-003-j*5.39240e-002, -1.25915e-002+j*5.38133e-003;
+2.43192e-001-j*3.17374e-002, -2.15730e-001-j*7.69941e-004, -1.56268e-001+j*1.44118e-002;
-7.27705e-002-j*3.54943e-003, -2.47706e-001+j*2.66480e-003, -1.21590e-001+j*1.06054e-002;
-7.25870e-002-j*4.62024e-003, +2.27073e-001-j*3.69315e-002, +1.22611e-001-j*6.67337e-003;
+2.32731e-001-j*2.85516e-002, +2.35389e-001-j*3.81905e-002, +5.35574e-002+j*4.30394e-004;
+2.64170e-001-j*2.67367e-002, -2.56227e-001+j*3.97957e-005, -1.95398e-001+j*2.23549e-002;
-1.66953e-002-j*7.95698e-003, -2.66547e-001-j*2.17687e-002, +1.56278e-002+j*2.23786e-003;
-3.42364e-002-j*9.30205e-003, +2.52340e-001-j*7.47237e-003, -9.51643e-004+j*3.64798e-003;
+2.97574e-001+j*0.00000e+000, +2.23170e-001-j*1.37831e-002, +1.06266e-001+j*2.30324e-003;
+2.67178e-002-j*4.15723e-004, +6.75423e-003-j*2.18428e-003, -1.69423e-002+j*3.12395e-003;
-1.12283e-002+j*2.86316e-004, +5.08225e-003-j*2.14053e-003, +2.18339e-002-j*3.25204e-003;
-1.17948e-002+j*6.82873e-004, +1.94914e-002-j*2.42151e-003, +2.68660e-003-j*2.92104e-004;
+1.19490e-002+j*1.72236e-005, +1.83552e-002-j*2.71289e-003, -2.70914e-002+j*4.84164e-003;
+1.00173e-002-j*5.80552e-005, -3.87262e-003-j*1.19607e-003, -8.53809e-003+j*1.48424e-003;
-1.22262e-002+j*5.13096e-004, -5.73905e-003-j*1.07659e-003, +3.51730e-002-j*6.13814e-003;
-1.43735e-002-j*4.78552e-004, +2.31135e-002-j*6.30554e-004, +1.80171e-003-j*1.98835e-004;
+9.17792e-003+j*5.36661e-004, +2.18969e-002-j*5.81759e-004, -3.72117e-002+j*5.35813e-003;
+3.38754e-002-j*3.38703e-002, -2.20843e-002+j*2.78581e-002, -8.79541e-002-j*3.67473e-003;
+3.93064e-002+j*4.69476e-002, -1.69132e-002-j*1.04606e-002, -1.85351e-002+j*1.33750e-003;
+3.60396e-002-j*2.46238e-002, +3.57722e-003+j*3.64827e-003, -1.92038e-002-j*6.65895e-002];

mode_shapes(:, :, 4) = [...
-1.02501e-001-j*1.43802e-001, -1.07971e-001+j*5.61418e-004, +1.87145e-001-j*1.03605e-001;
-9.44764e-002-j*1.36856e-001, -1.04428e-001+j*5.27790e-003, +1.60710e-001-j*7.74212e-002;
-9.11657e-002-j*1.36611e-001, -1.78165e-001-j*3.47193e-002, +2.37121e-001-j*4.96494e-002;
-9.17242e-002-j*1.36656e-001, -1.34249e-001-j*1.03884e-002, +1.92123e-001-j*1.25627e-001;
+1.05875e-002-j*1.03886e-001, -8.26338e-002+j*3.58498e-002, +2.55819e-001-j*6.94290e-003;
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mode_shapes(:, :, 5) = [...
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mode_shapes(:, :, 6) = [...
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mode_shapes(:, :, 7) = [...
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mode_shapes(:, :, 8) = [...
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mode_shapes(:, :, 9) = [...
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mode_shapes(:, :, 10) = [...
+3.33349e-002-j*4.89606e-003, -8.67138e-002-j*1.69402e-002, +2.87366e-002-j*1.66842e-002;
+2.95730e-002-j*6.10477e-004, -9.24590e-002-j*1.92562e-002, +5.21162e-002-j*1.31811e-002;
+3.26966e-002+j*1.03975e-002, -8.55682e-002-j*4.71847e-002, +3.99404e-002+j*8.59358e-003;
+3.45452e-002-j*3.05951e-003, -7.73823e-002-j*2.32199e-002, +2.30960e-002-j*7.49928e-003;
-3.72461e-003-j*5.40336e-003, +1.80151e-001-j*1.42898e-002, -8.63921e-003+j*1.95638e-002;
-1.11285e-002-j*2.22175e-003, +6.92355e-002-j*1.24144e-002, +8.02097e-002-j*6.81531e-003;
-2.99885e-002-j*4.21951e-004, +8.19709e-002+j*1.22484e-002, +3.97531e-002+j*1.33874e-002;
-1.59231e-002-j*3.96929e-003, +9.31295e-002+j*9.75532e-003, -4.01947e-002-j*5.17841e-003;
-7.32828e-003-j*2.93496e-002, +4.39909e-002+j*1.25298e-002, -8.87525e-002+j*2.05359e-002;
+8.20167e-002+j*6.86693e-004, +4.25475e-001-j*2.36494e-002, +3.37034e-001-j*2.44199e-002;
-2.68694e-002+j*3.82921e-003, +4.79292e-001+j*1.35903e-002, +5.06762e-001+j*0.00000e+000;
+1.40565e-002-j*5.41957e-003, +1.17563e-001+j*2.50398e-002, -5.97219e-002+j*5.92813e-004;
+2.30939e-002+j*1.16755e-002, +4.11136e-003+j*1.65726e-002, +2.41863e-002+j*5.06658e-002;
-4.92960e-003+j*9.24082e-003, -5.44667e-003+j*1.41983e-002, -3.73537e-002-j*1.87263e-002;
+9.68682e-003+j*1.11791e-002, -5.13436e-003+j*1.29205e-002, -4.88765e-002-j*4.97821e-002;
-1.66711e-003+j*1.03280e-002, +9.57955e-004+j*1.28350e-002, +3.30268e-002+j*5.44211e-002;
+5.90540e-003+j*1.67113e-002, +1.34280e-002+j*2.62111e-002, +4.38613e-002+j*7.17028e-002;
+4.97752e-003+j*1.47634e-002, +1.56773e-002+j*3.16026e-002, -5.27493e-002-j*3.38315e-002;
+2.83485e-003+j*1.14816e-002, +9.80676e-003+j*2.51504e-002, -6.78645e-002-j*9.90875e-002;
+5.62294e-003+j*1.84035e-002, +1.03717e-002+j*2.37801e-002, +2.58497e-002+j*3.75352e-002;
-8.75236e-003-j*4.71723e-003, +6.99107e-002+j*1.51894e-002, +3.46273e-002+j*1.48547e-002;
-1.72822e-002-j*8.75192e-003, +6.96759e-002+j*1.59783e-002, +8.07917e-002+j*1.97809e-002;
-1.29601e-002-j*5.61834e-003, +1.76126e-002+j*4.62761e-003, +6.64667e-002+j*1.61199e-002];

Define a point for each solid body

We define accelerometer indices used to define the motion of each solid body (2 3-axis accelerometer are enough).

  stages = [17, 19; % Bottom Granite
            13, 15; % Top Granite
            9,  11; % Ty
            5,  7;  % Ry
            21, 22; % Spindle
            1,  3]; % Hexapod

We define the origin point ${}^AO_B$ of the solid body $\{B\}$. Here we choose the middle point between the two accelerometers. This could be define differently (for instance by choosing the center of mass).

  AOB = zeros(3, size(stages, 1));

  for i = 1:size(stages, 1)
    AOB(:, i) = mean(positions(stages(i, :), 1:3))';
  end

Then we compute the positions of the sensors with respect to the previously defined origin for the frame $\{B\}$: ${}^BP_1$ and ${}^BP_2$.

  BP1 = zeros(3, size(stages, 1));
  BP2 = zeros(3, size(stages, 1));

  for i = 1:size(stages, 1)
    BP1(:, i) = positions(stages(i, 1), 1:3)' - AOB(:, i);
    BP2(:, i) = positions(stages(i, 2), 1:3)' - AOB(:, i);
  end

Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body. We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$.

We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^Av_{P_1}$ and ${}^Av_{P_2}$.

Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$.

Then we have:

\begin{align} {}^Av_{P_1} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_1 \\ {}^Av_{P_2} &= {}^Av_{O_B} + {}^A\Omega^\times {}^AR_B {}^BP_2 \end{align}

And we obtain

\begin{align} {}^A\Omega^\times {}^AR_B &= \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\ {}^Av_{O_B} &= {}^Av_{P_1} - \left( {}^Av_{P_2} - {}^Av_{P_1} \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1 \end{align}

  AVOB = zeros(3, size(stages, 1));
  ARB = zeros(3, 3, size(stages, 1));
  for i = 1:size(stages, 1)
    AVOB(:, i) = mode1(stages(i, 1), :)' - (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i))*BP1(:, i);
    ARB(:, :, i) = (mode1(stages(i, 2), :)' - mode1(stages(i, 1), :)')*pinv(BP2(:, i) - BP1(:, i));
  end

Argand Diagram

For mode 1

  figure;
  hold on;
  for i=1:size(mode1, 1)
    plot([0, real(mode1(i, 1))], [0, imag(mode1(i, 1))], '-k')
    plot([0, real(mode1(i, 2))], [0, imag(mode1(i, 2))], '-k')
    plot([0, real(mode1(i, 3))], [0, imag(mode1(i, 3))], '-k')

    % plot([0, real(mode2(i, 1))], [0, imag(mode2(i, 1))], '-r')
    % plot([0, real(mode2(i, 2))], [0, imag(mode2(i, 2))], '-r')
    % plot([0, real(mode2(i, 3))], [0, imag(mode2(i, 3))], '-r')

    % plot([0, real(mode3(i, 1))], [0, imag(mode3(i, 1))], '-b')
    % plot([0, real(mode3(i, 2))], [0, imag(mode3(i, 2))], '-b')
    % plot([0, real(mode3(i, 3))], [0, imag(mode3(i, 3))], '-b')
  end

  for i=1:size(AVOB, 2)
    plot([0, real(AVOB(1, i))], [0, imag(AVOB(1, i))], '-r')
    plot([0, real(AVOB(2, i))], [0, imag(AVOB(2, i))], '-r')
    plot([0, real(AVOB(3, i))], [0, imag(AVOB(3, i))], '-r')
  end

  % ang=0:0.01:2*pi;
  % radius1 = max(max(sqrt(real(mode1).^2+imag(mode1).^2)));
  % plot(radius1*cos(ang), radius1*sin(ang), '-k');

  % radius2 = max(max(sqrt(real(mode2).^2+imag(mode2).^2)));
  % plot(radius2*cos(ang), radius2*sin(ang), '-r');

  % radius3 = max(max(sqrt(real(mode3).^2+imag(mode3).^2)));
  % plot(radius3*cos(ang), radius3*sin(ang), '-b');

  hold off;
  axis manual equal

TEST: animate first mode

  figure;
  hold on;
  fill3(positions(1:4, 1),   positions(1:4, 2),   positions(1:4, 3),   'k', 'FaceAlpha', 0.5)
  fill3(positions(5:8, 1),   positions(5:8, 2),   positions(5:8, 3),   'k', 'FaceAlpha', 0.5)
  fill3(positions(9:12, 1),  positions(9:12, 2),  positions(9:12, 3),  'k', 'FaceAlpha', 0.5)
  fill3(positions(13:16, 1), positions(13:16, 2), positions(13:16, 3), 'k', 'FaceAlpha', 0.5)
  fill3(positions(17:20, 1), positions(17:20, 2), positions(17:20, 3), 'k', 'FaceAlpha', 0.5)
  fill3(positions(21:23, 1), positions(21:23, 2), positions(21:23, 3), 'k', 'FaceAlpha', 0.5)
  hold off;

  rec = polyshape([-2 -2 2 2],[-3 3 3 -3]);
  h = figure;
  filename = 'figs/mode_shapes.gif';
  n = 20;

  for i = 1:n
      axis manual equal

      Dm = real(V(1:3, 5)*cos(2*pi*i/n));
      rec_i = rotate(rec, 180/pi*Dm(3));
      rec_i = translate(rec_i, 10*Dm(1), 10*Dm(2));
      plot(rec_i);

      xlim([-3, 3]); ylim([-4, 4]);

      set(h, 'visible', 'off');
      set(h, 'pos', [0, 0, 500, 500]);
      drawnow;

      % Capture the plot as an image
      frame = getframe(h);
      im = frame2im(frame);
      [imind,cm] = rgb2ind(im,256);

      % Write to the GIF File
      if i == 1
          imwrite(imind,cm,filename,'gif','DelayTime',0.1,'Loopcount',inf);
      else
          imwrite(imind,cm,filename,'gif','DelayTime',0.1,'WriteMode','append');
      end
  end

  set(h, 'visible', 'on');
  ans = filename;

From 6 translations to translation + rotation

Let's define one absolute frame $\{A\}$ and one frame $\{B\}$ fixed w.r.t. the solid body. We note ${}^AO_B$ the position of origin of $\{B\}$ expressed in $\{A\}$.

We are measuring with the accelerometers the absolute motion of points $P_1$ and $P_2$: ${}^AP_1$ and ${}^AP_2$.

Let's note ${}^BP_1$ and ${}^BP_2$ the (known) coordinates of $P_1$ and $P_2$ expressed in the frame $\{B\}$.

Then we have:

\begin{align} {}^AP_1 &= {}^AO_B + {}^AR_B {}^BP_1 \\ {}^AP_2 &= {}^AO_B + {}^AR_B {}^BP_2 \end{align}

And we obtain

\begin{align} {}^AR_B &= \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1}\\ {}^AO_B &= {}^Av_{P_1} - \left( {}^AP_2 - {}^AP_1 \right) \left( {}^BP_2 - {}^BP_1 \right)^{-1} {}^BP_1 \end{align}

Problem with AirLoc System

4 Airloc Levelers are used for the granite (figure fig:airloc).

AirLoc used for the granite (2120-KSKC)

They are probably not well leveled so that could explain the first modes at 11Hz and 17Hz.

Spatial Mode Extraction