242 lines
7.3 KiB
Org Mode
242 lines
7.3 KiB
Org Mode
#+TITLE: Identification of the Stewart Platform using Simscape
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:DRAWER:
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#+STARTUP: overview
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ./index.html
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#+HTML_LINK_UP: ./index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="./css/readtheorg.css"/>
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#+HTML_HEAD: <script src="./js/jquery.min.js"></script>
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#+HTML_HEAD: <script src="./js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script src="./js/jquery.stickytableheaders.min.js"></script>
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#+HTML_HEAD: <script src="./js/readtheorg.js"></script>
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+PROPERTY: header-args:matlab+ :results none
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#+PROPERTY: header-args:matlab+ :eval no-export
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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* Introduction :ignore:
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* Modal Analysis of the Stewart Platform
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :results none :exports none
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simulinkproject('../');
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#+end_src
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#+begin_src matlab
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open('stewart_platform_model.slx')
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#+end_src
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** Initialize the Stewart Platform
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#+begin_src matlab
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stewart = initializeStewartPlatform();
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stewart = initializeFramesPositions(stewart);
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stewart = generateGeneralConfiguration(stewart);
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stewart = computeJointsPose(stewart);
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stewart = initializeStrutDynamics(stewart);
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stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
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stewart = initializeCylindricalPlatforms(stewart);
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stewart = initializeCylindricalStruts(stewart);
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stewart = computeJacobian(stewart);
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stewart = initializeStewartPose(stewart);
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stewart = initializeInertialSensor(stewart);
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#+end_src
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#+begin_src matlab
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ground = initializeGround('type', 'none');
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payload = initializePayload('type', 'none');
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#+end_src
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** Identification
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#+begin_src matlab
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%% Options for Linearized
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options = linearizeOptions;
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options.SampleTime = 0;
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%% Name of the Simulink File
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mdl = 'stewart_platform_model';
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%% Input/Output definition
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clear io; io_i = 1;
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io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N]
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Position/Orientation of {B} w.r.t. {A}
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io(io_i) = linio([mdl, '/Relative Motion Sensor'], 2, 'openoutput'); io_i = io_i + 1; % Velocity of {B} w.r.t. {A}
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%% Run the linearization
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G = linearize(mdl, io);
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% G.InputName = {'tau1', 'tau2', 'tau3', 'tau4', 'tau5', 'tau6'};
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% G.OutputName = {'Xdx', 'Xdy', 'Xdz', 'Xrx', 'Xry', 'Xrz', 'Vdx', 'Vdy', 'Vdz', 'Vrx', 'Vry', 'Vrz'};
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#+end_src
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Let's check the size of =G=:
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#+begin_src matlab :results replace output
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size(G)
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#+end_src
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#+RESULTS:
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: size(G)
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: State-space model with 12 outputs, 6 inputs, and 18 states.
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: 'org_babel_eoe'
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: ans =
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: 'org_babel_eoe'
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We expect to have only 12 states (corresponding to the 6dof of the mobile platform).
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#+begin_src matlab :results replace output
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Gm = minreal(G);
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#+end_src
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#+RESULTS:
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: Gm = minreal(G);
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: 6 states removed.
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And indeed, we obtain 12 states.
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** Coordinate transformation
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We can perform the following transformation using the =ss2ss= command.
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#+begin_src matlab
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Gt = ss2ss(Gm, Gm.C);
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#+end_src
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Then, the =C= matrix of =Gt= is the unity matrix which means that the states of the state space model are equal to the measurements $\bm{Y}$.
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The measurements are the 6 displacement and 6 velocities of mobile platform with respect to $\{B\}$.
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We could perform the transformation by hand:
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#+begin_src matlab
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At = Gm.C*Gm.A*pinv(Gm.C);
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Bt = Gm.C*Gm.B;
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Ct = eye(12);
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Dt = zeros(12, 6);
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Gt = ss(At, Bt, Ct, Dt);
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#+end_src
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** Analysis
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#+begin_src matlab
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[V,D] = eig(Gt.A);
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no :post addhdr(*this*)
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ws = imag(diag(D))/2/pi;
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[ws,I] = sort(ws)
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xi = 100*real(diag(D))./imag(diag(D));
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xi = xi(I);
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data2orgtable([[1:length(ws(ws>0))]', ws(ws>0), xi(xi>0)], {}, {'Mode Number', 'Resonance Frequency [Hz]', 'Damping Ratio [%]'}, ' %.1f ');
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#+end_src
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#+RESULTS:
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| Mode Number | Resonance Frequency [Hz] | Damping Ratio [%] |
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|-------------+--------------------------+-------------------|
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| 1.0 | 780.6 | 0.4 |
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| 2.0 | 780.6 | 0.3 |
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| 3.0 | 903.9 | 0.3 |
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| 4.0 | 1061.4 | 0.3 |
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| 5.0 | 1061.4 | 0.2 |
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| 6.0 | 1269.6 | 0.2 |
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** Visualizing the modes
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To visualize the i'th mode, we may excite the system using the inputs $U_i$ such that $B U_i$ is co-linear to $\xi_i$ (the mode we want to excite).
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\[ U(t) = e^{\alpha t} ( ) \]
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Let's first sort the modes and just take the modes corresponding to a eigenvalue with a positive imaginary part.
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#+begin_src matlab
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ws = imag(diag(D));
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[ws,I] = sort(ws)
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ws = ws(7:end); I = I(7:end);
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#+end_src
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#+begin_src matlab
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for i = 1:length(ws)
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#+end_src
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#+begin_src matlab
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i_mode = I(i); % the argument is the i'th mode
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#+end_src
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#+begin_src matlab
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lambda_i = D(i_mode, i_mode);
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xi_i = V(:,i_mode);
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a_i = real(lambda_i);
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b_i = imag(lambda_i);
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#+end_src
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Let do 10 periods of the mode.
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#+begin_src matlab
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t = linspace(0, 10/(imag(lambda_i)/2/pi), 1000);
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U_i = pinv(Gt.B) * real(xi_i * lambda_i * (cos(b_i * t) + 1i*sin(b_i * t)));
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#+end_src
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#+begin_src matlab
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U = timeseries(U_i, t);
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#+end_src
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Simulation:
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#+begin_src matlab
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load('mat/conf_simscape.mat');
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set_param(conf_simscape, 'StopTime', num2str(t(end)));
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sim(mdl);
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#+end_src
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Save the movie of the mode shape.
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#+begin_src matlab
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smwritevideo(mdl, sprintf('figs/mode%i', i), ...
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'PlaybackSpeedRatio', 1/(b_i/2/pi), ...
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'FrameRate', 30, ...
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'FrameSize', [800, 400]);
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#+end_src
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#+begin_src matlab
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end
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#+end_src
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#+name: fig:mode1
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#+caption: Identified mode - 1
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[[file:figs/mode1.gif]]
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#+name: fig:mode3
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#+caption: Identified mode - 3
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[[file:figs/mode3.gif]]
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#+name: fig:mode5
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#+caption: Identified mode - 5
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[[file:figs/mode5.gif]]
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