%% Clear Workspace and Close figures clear; close all; clc; %% Intialize Laplace variable s = zpk('s'); % Run meas_transformation.m script in order to get the tranformation (jacobian) matrix run('meas_transformation.m') % Open the Simulink File open('vibration_table') % Springs % <> % The 4 springs supporting the suspended optical table are modelled with "bushing joints" having stiffness and damping in the x-y-z directions: spring.kx = 1e4; % X- Stiffness [N/m] spring.cx = 1e1; % X- Damping [N/(m/s)] spring.ky = 1e4; % Y- Stiffness [N/m] spring.cy = 1e1; % Y- Damping [N/(m/s)] spring.kz = 1e4; % Z- Stiffness [N/m] spring.cz = 1e1; % Z- Damping [N/(m/s)] spring.z0 = 32e-3; % Equilibrium z-length [m] % Inertial Shaker (IS20) % <> % The inertial shaker is defined as two solid bodies: % - the "housing" that is fixed to the element that we want to excite % - the "inertial mass" that is suspended inside the housing % The inertial mass is guided inside the housing and an actuator (coil and magnet) can be used to apply a force between the inertial mass and the support. % The "reacting" force on the support is then used as an excitation. % #+name: tab:is20_characteristics % #+caption: Summary of the IS20 datasheet % #+attr_latex: :environment tabularx :width 0.4\linewidth :align lX % #+attr_latex: :center t :booktabs t :float t % | Characteristic | Value | % |-----------------+------------| % | Output Force | 20 N | % | Frequency Range | 10-3000 Hz | % | Moving Mass | 0.1 kg | % | Total Mass | 0.3 kg | % From the datasheet in Table [[tab:is20_characteristics]], we can estimate the parameters of the physical shaker. % These parameters are defined below shaker.w0 = 2*pi*10; % Resonance frequency of moving mass [rad/s] shaker.m = 0.1; % Moving mass [m] shaker.m_tot = 0.3; % Total mass [m] shaker.k = shaker.m*shaker.w0^2; % Spring constant [N/m] shaker.c = 0.2*sqrt(shaker.k*shaker.m); % Damping [N/(m/s)] % 3D accelerometer (356B18) % <> % An accelerometer consists of 2 solids: % - a "housing" rigidly fixed to the measured body % - an "inertial mass" suspended inside the housing by springs and guided in the measured direction % 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). % #+name: tab:356b18_characteristics % #+caption: Summary of the 356B18 datasheet % #+attr_latex: :environment tabularx :width 0.5\linewidth :align lX % #+attr_latex: :center t :booktabs t :float t % | Characteristic | Value | % |---------------------+---------------------| % | Sensitivity | 0.102 V/(m/s2) | % | Frequency Range | 0.5 to 3000 Hz | % | Resonance Frequency | > 20 kHz | % | Resolution | 0.0005 m/s2 rms | % | Weight | 0.025 kg | % | Size | 20.3x26.1x20.3 [mm] | % Here are defined the parameters for the triaxial accelerometer: acc_3d.m = 0.005; % Inertial mass [kg] acc_3d.m_tot = 0.025; % Total mass [m] acc_3d.w0 = 2*pi*20e3; % Resonance frequency [rad/s] acc_3d.kx = acc_3d.m*acc_3d.w0^2; % Spring constant [N/m] acc_3d.ky = acc_3d.m*acc_3d.w0^2; % Spring constant [N/m] acc_3d.kz = acc_3d.m*acc_3d.w0^2; % Spring constant [N/m] acc_3d.cx = 1e2; % Damping [N/(m/s)] acc_3d.cy = 1e2; % Damping [N/(m/s)] acc_3d.cz = 1e2; % Damping [N/(m/s)] % DC gain between support acceleration and inertial mass displacement is $-m/k$: acc_3d.g_x = 1/(-acc_3d.m/acc_3d.kx); % [m/s^2/m] acc_3d.g_y = 1/(-acc_3d.m/acc_3d.ky); % [m/s^2/m] acc_3d.g_z = 1/(-acc_3d.m/acc_3d.kz); % [m/s^2/m] % We also define the sensitivity in order to have the outputs in volts. acc_3d.gV_x = 0.102; % [V/(m/s^2)] acc_3d.gV_y = 0.102; % [V/(m/s^2)] acc_3d.gV_z = 0.102; % [V/(m/s^2)] % The problem with using such model for accelerometers is that this adds states to the identified models (2x3 states for each triaxial accelerometer). % These states represents the dynamics of the suspended inertial mass. % In the frequency band of interest (few Hz up to ~1 kHz), the dynamics of the inertial mass can be ignore (its resonance is way above 1kHz). % Therefore, we might as well use idealized "transform sensors" blocks as they will give the same result up to ~20kHz while allowing to reduce the number of identified states. % The accelerometer model can be chosen by setting the =type= property: acc_3d.type = 2; % 1: inertial mass, 2: perfect % Number of states % Let's first use perfect 3d accelerometers: acc_3d.type = 2; % 1: inertial mass, 2: perfect % And identify the dynamics from the shaker force to the measured accelerations: %% Name of the Simulink File mdl = 'vibration_table'; %% Input/Output definition 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; %% Run the linearization Gp = linearize(mdl, io); Gp.InputName = {'F'}; Gp.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6'}; % #+RESULTS: % : size(Gp) % : State-space model with 6 outputs, 1 inputs, and 12 states. % We indeed have the 12 states corresponding to the 6 DoF of the suspended optical table. % Let's now consider the inertial masses for the triaxial accelerometers: acc_3d.type = 1; % 1: inertial mass, 2: perfect %% Name of the Simulink File mdl = 'vibration_table'; %% Input/Output definition 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; %% Run the linearization Ga = linearize(mdl, io); Ga.InputName = {'F'}; Ga.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6'}; % Resonance frequencies and mode shapes % Let's now identify the resonance frequency and mode shapes associated with the suspension modes of the optical table. acc_3d.type = 2; % 1: inertial mass, 2: perfect %% Name of the Simulink File mdl = 'vibration_table'; %% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/F'], 1, 'openinput'); 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); G.InputName = {'F'}; G.OutputName = {'ax', 'ay', 'az', 'wx', 'wy', 'wz'}; % Compute the resonance frequencies ws = eig(G.A); ws = ws(imag(ws) > 0); % And the associated response of the optical table x_mod = zeros(6, 6); for i = 1:length(ws) xi = evalfr(G, ws(i)); x_mod(:,i) = xi./norm(xi); end % Verify transformation %% Options for Linearized options = linearizeOptions; options.SampleTime = 0; %% Name of the Simulink File mdl = 'vibration_table'; %% Input/Output definition 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, '/acc_O'], 1, 'openoutput'); io_i = io_i + 1; %% Run the linearization G = linearize(mdl, io, 0.0, options); G.InputName = {'F'}; G.OutputName = {'a1', 'a2', 'a3', 'a4', 'a5', 'a6', ... 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; G_acc = inv(M)*G(1:6, 1); G_id = G(7:12, 1); bodeFig({G_acc(1), G_id(1)}) bodeFig({G_acc(2), G_id(2)}) bodeFig({G_acc(3), G_id(3)}) bodeFig({G_acc(4), G_id(4)}) bodeFig({G_acc(5), G_id(5)}) bodeFig({G_acc(6), G_id(6)})