diff --git a/matlab/R2019B/accelerometer_3d.slx b/matlab/R2019B/accelerometer_3d.slx index fa71675..72bce52 100644 Binary files a/matlab/R2019B/accelerometer_3d.slx and b/matlab/R2019B/accelerometer_3d.slx differ diff --git a/matlab/R2019B/accelerometer_3d_perfect.slx b/matlab/R2019B/accelerometer_3d_perfect.slx index 867f7c8..0945a76 100644 Binary files a/matlab/R2019B/accelerometer_3d_perfect.slx and b/matlab/R2019B/accelerometer_3d_perfect.slx differ diff --git a/matlab/R2019B/vibration_table.slx b/matlab/R2019B/vibration_table.slx index 0cbc848..cedb862 100644 Binary files a/matlab/R2019B/vibration_table.slx and b/matlab/R2019B/vibration_table.slx differ diff --git a/matlab/vibration_table.slx b/matlab/vibration_table.slx index 0f38b28..a8ef67f 100644 Binary files a/matlab/vibration_table.slx and b/matlab/vibration_table.slx differ diff --git a/vibration-table.html b/vibration-table.html index dba3831..459f52c 100644 --- a/vibration-table.html +++ b/vibration-table.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Vibration Table @@ -39,41 +39,41 @@

Table of Contents

@@ -81,33 +81,33 @@

This report is also available as a pdf.

-
-

1 Introduction

+
+

1 Introduction

This document is divided as follows:

    -
  • Section 2: the experimental setup and all the instrumentation are described
    • -
  • Section 3: the mathematics used to compute the 6DoF motion of a solid body from several inertial sensor is derived
    • -
  • Section 4: a Simscape model of the vibration table is developed
    • -
  • Section 5: the table dynamics is identified and compared with the Simscape model
    • +
  • Section 2: the experimental setup and all the instrumentation are described
    • +
  • Section 3: the mathematics used to compute the 6DoF motion of a solid body from several inertial sensor is derived
    • +
  • Section 4: a Simscape model of the vibration table is developed
    • +
  • Section 5: the table dynamics is identified and compared with the Simscape model
-
-

2 Experimental Setup

+
+

2 Experimental Setup

- +

-
-

2.1 CAD Model

+
+

2.1 CAD Model

-
+

vibration-table-cad-view.png

Figure 1: CAD View of the vibration table

@@ -115,10 +115,10 @@ This document is divided as follows:
-
-

2.2 Instrumentation

+
+

2.2 Instrumentation

-
+

Here are the documentation of the equipment used for this vibration table:

@@ -136,8 +136,8 @@ Here are the documentation of the equipment used for this vibration table:
-
-

2.3 Suspended table

+
+

2.3 Suspended table

Dimensions
450 mm x 450 mm x 60 mm
@@ -145,7 +145,7 @@ Here are the documentation of the equipment used for this vibration table:
-
+

B4545A_Compliance_inLb-780.png

Figure 2: Compliance of the B4545A optical table

@@ -153,8 +153,8 @@ Here are the documentation of the equipment used for this vibration table:
-
-

2.4 Inertial Sensors

+
+

2.4 Inertial Sensors

@@ -181,18 +181,18 @@ Here are the documentation of the equipment used for this vibration table: -
-

3 Compute the 6DoF solid body motion from several inertial sensors

+
+

3 Compute the 6DoF solid body motion from several inertial sensors

- +

-Let’s consider a solid body with several accelerometers attached to it (Figure 3). +Let’s consider a solid body with several accelerometers attached to it (Figure 3).

-
+

local_to_global_coordinates.png

Figure 3: Schematic of the measured motions of a solid body

@@ -222,7 +222,7 @@ The measurement of the individual vectors is defined as the vector \(\vec{a}\): \end{equation}

-From the positions and orientations of the acceleremoters (defined in Section 3.1), it is quite straightforward to compute the accelerations measured by the sensors from the acceleration/angular acceleration of the solid body (Section 3.2). +From the positions and orientations of the acceleremoters (defined in Section 3.1), it is quite straightforward to compute the accelerations measured by the sensors from the acceleration/angular acceleration of the solid body (Section 3.2). From this, we can easily build a transformation matrix \(M\), such that:

\begin{equation} @@ -230,7 +230,7 @@ From this, we can easily build a transformation matrix \(M\), such that: \end{equation}

-If the matrix is invertible, we can just take the inverse in order to obtain the transformation matrix giving the 6dof acceleration of the solid body from the accelerometer measurements (Section 3.3): +If the matrix is invertible, we can just take the inverse in order to obtain the transformation matrix giving the 6dof acceleration of the solid body from the accelerometer measurements (Section 3.3):

\begin{equation} {}^O\vec{x} = M^{-1} \cdot \vec{a} @@ -241,11 +241,11 @@ If it is not invertible, then it means that it is not possible to compute all 6d The solution is then to change the location/orientation of some of the accelerometers.

-
-

3.1 Define accelerometers positions/orientations

+
+

3.1 Define accelerometers positions/orientations

- + Let’s first define the position and orientation of all measured accelerations with respect to a defined frame \(\{O\}\).

@@ -260,10 +260,10 @@ Let’s first define the position and orientation of all measured accelerati

-There are summarized in Table 1. +There are summarized in Table 1.

-
+
@@ -339,10 +339,10 @@ We then define the direction of the measured accelerations (unit vectors):

-They are summarized in Table 2. +They are summarized in Table 2.

-
Table 1: Positions of the accelerometers fixed to the vibration table with respect to \(\{O\}\)
+
@@ -406,11 +406,11 @@ They are summarized in Table 2. -
-

3.2 Transformation matrix from motion of the solid body to accelerometer measurements

+
+

3.2 Transformation matrix from motion of the solid body to accelerometer measurements

- +

@@ -475,7 +475,7 @@ a_i = \begin{bmatrix} And finally we can combine the 6 (line) vectors for the 6 accelerometers to write that in a matrix form. We obtain Eq. \eqref{eq:M_matrix}.

-
+

The transformation from solid body acceleration \({}^O\vec{x}\) from sensor measured acceleration \(\vec{a}\) is:

@@ -510,10 +510,10 @@ Let’s define such matrix using matlab:

-The obtained matrix is shown in Table 3. +The obtained matrix is shown in Table 3.

-
Table 2: Orientations of the accelerometers fixed to the vibration table expressed in \(\{O\}\)
+
@@ -607,11 +607,11 @@ The obtained matrix is shown in Table 3. -
-

3.3 Compute the transformation matrix from accelerometer measurement to motion of the solid body

+
+

3.3 Compute the transformation matrix from accelerometer measurement to motion of the solid body

- +

@@ -630,10 +630,10 @@ We therefore need the determinant of \(M\) to be non zero:

-The obtained inverse of the matrix is shown in Table 4. +The obtained inverse of the matrix is shown in Table 4.

-
Table 3: Effect of a displacement/rotation on the 6 measurements
+
@@ -728,28 +728,28 @@ The obtained inverse of the matrix is shown in Table 4 -
-

4 Simscape Model

+
+

4 Simscape Model

- +

In this section, the Simscape model of the vibration table is described.

-
+

simscape_vibration_table.png

Figure 4: 3D representation of the simscape model

-
-

4.1 Simscape Sub-systems

+
+

4.1 Simscape Sub-systems

- +

@@ -757,11 +757,11 @@ Parameters for sub-components of the simscape model are defined below.

-
-

4.1.1 Springs

+
+

4.1.1 Springs

- +

@@ -781,14 +781,22 @@ spring.cz = 1e1; % Z- Damping [N/(m/s)] spring.z0 = 32e-3; % Equilibrium z-length [m]

+ +

+And we can increase the “equilibrium position” of the vertical springs to take into account the gravity forces and start closer to equilibrium: +

+
+
spring.dl = (30.5918/4)*9.80665/spring.kz;
+
+
-
-

4.1.2 Inertial Shaker (IS20)

+
+

4.1.2 Inertial Shaker (IS20)

- +

@@ -804,7 +812,7 @@ The inertial mass is guided inside the housing and an actuator (coil and magnet) The “reacting” force on the support is then used as an excitation.

-
Table 4: Compute the displacement/rotation from the 6 measurements
+
@@ -842,7 +850,7 @@ The “reacting” force on the support is then used as an excitation.
Table 5: Summary of the IS20 datasheet

-From the datasheet in Table 5, we can estimate the parameters of the physical shaker. +From the datasheet in Table 5, we can estimate the parameters of the physical shaker.

@@ -859,11 +867,11 @@ shaker.c = 0.2*sqrt(shaker.k

-
-

4.1.3 3D accelerometer (356B18)

+
+

4.1.3 3D accelerometer (356B18)

- +

@@ -878,7 +886,7 @@ An accelerometer consists of 2 solids: The relative motion between the housing and the inertial mass gives a measurement of the acceleration of the measured body (up to the suspension mode of the inertial mass).

- +
@@ -982,16 +990,16 @@ The accelerometer model can be chosen by setting the type property: -
-

4.2 Identification

+
+

4.2 Identification

- +

-
-

4.2.1 Number of states

+
+

4.2.1 Number of states

Let’s first use perfect 3d accelerometers: @@ -1067,8 +1075,8 @@ This corresponds to 6 states for each triaxial accelerometers.

-
-

4.2.2 Resonance frequencies and mode shapes

+
+

4.2.2 Resonance frequencies and mode shapes

Let’s now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table. @@ -1115,11 +1123,11 @@ And the associated response of the optical table

-The results are shown in Table 7. +The results are shown in Table 7. The motion associated to the mode shapes are just indicative.

-
Table 6: Summary of the 356B18 datasheet
+
@@ -1214,8 +1222,8 @@ The motion associated to the mode shapes are just indicative. -
-

4.3 Verify transformation

+
+

4.3 Verify transformation

%% Options for Linearized
@@ -1229,7 +1237,7 @@ mdl = 'vibration_table';
 clear io; io_i = 1;
 io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1;
 io(io_i) = linio([mdl, '/acc'],  1, 'openoutput'); io_i = io_i + 1;
-io(io_i) = linio([mdl, '/Absolute_Accelerometer'],  1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/acc_O'],  1, 'openoutput'); io_i = io_i + 1;
 
 %% Run the linearization
 G = linearize(mdl, io, 0.0, options);
@@ -1258,18 +1266,18 @@ bodeFig({G_acc(6), G_id(6)})
-
-

5 Identification of the table’s dynamics

+
+

5 Identification of the table’s dynamics

- +

Author: Dehaeze Thomas

-

Created: 2021-04-16 ven. 18:28

+

Created: 2021-04-19 lun. 15:02

diff --git a/vibration-table.org b/vibration-table.org index c8b19fc..8bc88be 100644 --- a/vibration-table.org +++ b/vibration-table.org @@ -456,6 +456,11 @@ spring.cz = 1e1; % Z- Damping [N/(m/s)] spring.z0 = 32e-3; % Equilibrium z-length [m] #+end_src +And we can increase the "equilibrium position" of the vertical springs to take into account the gravity forces and start closer to equilibrium: +#+begin_src matlab +spring.dl = (30.5918/4)*9.80665/spring.kz; +#+end_src + *** Inertial Shaker (IS20) <> diff --git a/vibration-table.pdf b/vibration-table.pdf index 4dc6fce..b107f9b 100644 Binary files a/vibration-table.pdf and b/vibration-table.pdf differ
Table 7: Resonance frequency and approximation of the mode shapes