Move accelerometer/geophone explaination
This commit is contained in:
@@ -301,106 +301,3 @@ This Matlab function is accessible [[file:../src/initializeGround.m][here]].
|
||||
ground.C = args.C;
|
||||
#+end_src
|
||||
|
||||
** Z-Axis Geophone
|
||||
*** Working Principle
|
||||
From the schematic of the Z-axis geophone shown in Figure [[fig:z_axis_geophone]], we can write the transfer function from the support velocity $\dot{w}$ to the relative velocity of the inertial mass $\dot{d}$:
|
||||
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
||||
with:
|
||||
- $\omega_0 = \sqrt{\frac{k}{m}}$
|
||||
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$
|
||||
|
||||
#+name: fig:z_axis_geophone
|
||||
#+caption: Schematic of a Z-Axis geophone
|
||||
[[file:figs/inertial_sensor.png]]
|
||||
|
||||
We see that at frequencies above $\omega_0$:
|
||||
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]
|
||||
|
||||
And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.
|
||||
|
||||
We generally want to have the smallest resonant frequency $\omega_0$ to measure low frequency absolute velocity, however there is a trade-off between $\omega_0$ and the mass of the inertial mass.
|
||||
|
||||
*** Initialization function
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle ../src/initializeZAxisGeophone.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:initializeZAxisGeophone>>
|
||||
|
||||
This Matlab function is accessible [[file:../src/initializeZAxisGeophone.m][here]].
|
||||
|
||||
#+begin_src matlab
|
||||
function [geophone] = initializeZAxisGeophone(args)
|
||||
arguments
|
||||
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % [kg]
|
||||
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 1 % [Hz]
|
||||
end
|
||||
|
||||
%%
|
||||
geophone.m = args.mass;
|
||||
|
||||
%% The Stiffness is set to have the damping resonance frequency
|
||||
geophone.k = geophone.m * (2*pi*args.freq)^2;
|
||||
|
||||
%% We set the damping value to have critical damping
|
||||
geophone.c = 2*sqrt(geophone.m * geophone.k);
|
||||
|
||||
%% Save
|
||||
save('./mat/geophone_z_axis.mat', 'geophone');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
** Z-Axis Accelerometer
|
||||
*** Working Principle
|
||||
From the schematic of the Z-axis accelerometer shown in Figure [[fig:z_axis_accelerometer]], we can write the transfer function from the support acceleration $\ddot{w}$ to the relative position of the inertial mass $d$:
|
||||
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
||||
with:
|
||||
- $\omega_0 = \sqrt{\frac{k}{m}}$
|
||||
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$
|
||||
|
||||
#+name: fig:z_axis_accelerometer
|
||||
#+caption: Schematic of a Z-Axis geophone
|
||||
[[file:figs/inertial_sensor.png]]
|
||||
|
||||
We see that at frequencies below $\omega_0$:
|
||||
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]
|
||||
|
||||
And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.
|
||||
|
||||
Note that there is trade-off between:
|
||||
- the highest measurable acceleration $\omega_0$
|
||||
- the sensitivity of the accelerometer which is equal to $-\frac{1}{{\omega_0}^2}$
|
||||
|
||||
*** Initialization function
|
||||
:PROPERTIES:
|
||||
:header-args:matlab+: :tangle ../src/initializeZAxisAccelerometer.m
|
||||
:header-args:matlab+: :comments none :mkdirp yes :eval no
|
||||
:END:
|
||||
<<sec:initializeZAxisAccelerometer>>
|
||||
|
||||
This Matlab function is accessible [[file:../src/initializeZAxisAccelerometer.m][here]].
|
||||
|
||||
#+begin_src matlab
|
||||
function [accelerometer] = initializeZAxisAccelerometer(args)
|
||||
arguments
|
||||
args.mass (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 % [kg]
|
||||
args.freq (1,1) double {mustBeNumeric, mustBePositive} = 5e3 % [Hz]
|
||||
end
|
||||
|
||||
%%
|
||||
accelerometer.m = args.mass;
|
||||
|
||||
%% The Stiffness is set to have the damping resonance frequency
|
||||
accelerometer.k = accelerometer.m * (2*pi*args.freq)^2;
|
||||
|
||||
%% We set the damping value to have critical damping
|
||||
accelerometer.c = 2*sqrt(accelerometer.m * accelerometer.k);
|
||||
|
||||
%% Gain correction of the accelerometer to have a unity gain until the resonance
|
||||
accelerometer.gain = -accelerometer.k/accelerometer.m;
|
||||
|
||||
%% Save
|
||||
save('./mat/accelerometer_z_axis.mat', 'accelerometer');
|
||||
end
|
||||
#+end_src
|
||||
|
||||
|
||||
@@ -1344,6 +1344,50 @@ Rotational Damping
|
||||
|
||||
This Matlab function is accessible [[file:../src/initializeInertialSensor.m][here]].
|
||||
|
||||
*** Geophone - Working Principle
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
From the schematic of the Z-axis geophone shown in Figure [[fig:z_axis_geophone]], we can write the transfer function from the support velocity $\dot{w}$ to the relative velocity of the inertial mass $\dot{d}$:
|
||||
\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
||||
with:
|
||||
- $\omega_0 = \sqrt{\frac{k}{m}}$
|
||||
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$
|
||||
|
||||
#+name: fig:z_axis_geophone
|
||||
#+caption: Schematic of a Z-Axis geophone
|
||||
[[file:figs/inertial_sensor.png]]
|
||||
|
||||
We see that at frequencies above $\omega_0$:
|
||||
\[ \frac{\dot{d}}{\dot{w}} \approx -1 \]
|
||||
|
||||
And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support.
|
||||
|
||||
We generally want to have the smallest resonant frequency $\omega_0$ to measure low frequency absolute velocity, however there is a trade-off between $\omega_0$ and the mass of the inertial mass.
|
||||
|
||||
*** Accelerometer - Working Principle
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
From the schematic of the Z-axis accelerometer shown in Figure [[fig:z_axis_accelerometer]], we can write the transfer function from the support acceleration $\ddot{w}$ to the relative position of the inertial mass $d$:
|
||||
\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \]
|
||||
with:
|
||||
- $\omega_0 = \sqrt{\frac{k}{m}}$
|
||||
- $\xi = \frac{1}{2} \sqrt{\frac{m}{k}}$
|
||||
|
||||
#+name: fig:z_axis_accelerometer
|
||||
#+caption: Schematic of a Z-Axis geophone
|
||||
[[file:figs/inertial_sensor.png]]
|
||||
|
||||
We see that at frequencies below $\omega_0$:
|
||||
\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \]
|
||||
|
||||
And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support.
|
||||
|
||||
Note that there is trade-off between:
|
||||
- the highest measurable acceleration $\omega_0$
|
||||
- the sensitivity of the accelerometer which is equal to $-\frac{1}{{\omega_0}^2}$
|
||||
|
||||
*** Function description
|
||||
:PROPERTIES:
|
||||
:UNNUMBERED: t
|
||||
@@ -1744,6 +1788,8 @@ Plot the legs connecting the joints of the fixed base to the joints of the mobil
|
||||
axis equal;
|
||||
axis off;
|
||||
title('Side')
|
||||
|
||||
close(f);
|
||||
end
|
||||
#+end_src
|
||||
|
||||
|
||||
Reference in New Issue
Block a user