Example for stick slip friction model

toolbox/examples/friction_model/friction_RHS_filippov.m

@@ -0,0 +1,16 @@
+function dx = friction_RHS_filippov(t, x, p)
+% p = (k, m, mu_b, F_s, delta, epsilon)
+k       = p(1);
+m       = p(2);
+mu_b    = p(3);
+F_s     = p(4);
+delta   = p(5);
+epsilon = p(6);
+mu_rel  = x(2) - mu_b;
+
+dx = zeros(2,1);
+dx(1) = x(2);
+
+dx(2) = -(k / m) * x(1) + F_fric_fil(x, mu_rel, epsilon, F_s, k, delta) / m;
+
+end

toolbox/examples/friction_model/friction_RHS_no_filippov.m

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+function dx = friction_RHS_no_filippov(t, x, p)
+% p = (k, m, mu_b, F_s, delta, epsilon)
+k       = p(1);
+m       = p(2);
+mu_b    = p(3);
+F_s     = p(4);
+delta   = p(5);
+epsilon = p(6);
+mu_rel  = x(2) - mu_b;
+
+dx = zeros(2,1);
+dx(1) = x(2);
+dx(2) = -(k / m) * x(1) + F_fric_no_fil(x, mu_rel, epsilon, F_s, k, delta) / m;
+
+end

toolbox/examples/friction_model/friction_model_comparison.m

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+%% Source: Stick-Slip Vibrations Induced by Alternate Friction Models
+% paper by: R. Leine, D. van Campen, A. de Kraker, and L. van den Steen
+% doi: 10.1023/A:1008289604683
+
+% The system describes dry friction of a spring suspended weight sitting on a 
+% constantly moving conveyor belt. The weight moves along with the belt
+% until spring tension becomes too high and the weight is pushed back.
+% This creates a switched periodic motion. 
+
+% The corresponding ODE can be formulated in a filippov and non-filippov
+% variant and serves as a perfect example to test the accuracy of not only
+% the filippov integration but also the resulting sensitivities.
+
+% Both formulations can be found in "Numerical Solution of Optimal Control 
+% Problems with Explicit and Implicit Switches" by Andreas Meyer
+% chapter: 15.3
+
+integrator = @ode45;
+t0 = 0;
+tf = 30;
+timeinterval = [t0,tf];
+initstates   = [1.133944669704  0 ];
+p(1) = 1.0;   %k
+p(2) = 1.0;   %m
+p(3) = 0.2;   %mu_b
+p(4) = 1;     %F_s
+p(5) = 3;     %delta
+epsilon = 1e-11;
+p(6) = epsilon; %epsilon
+
+% saving current warning state for filippov detection (then turn off warning)
+initial_filippov_warning_state = warning('query', 'IFDIFF:chattering').state;
+warning('off', 'IFDIFF:chattering');
+
+%% Solving the friction model variant without filippov mode
+% preprocessing
+fprintf('Preprocessing...\n  ');
+odeoptions = odeset( 'AbsTol', 1e-20, 'RelTol', 1e-6);
+filename = 'friction_RHS_no_filippov';
+dhandle = prepareDatahandleForIntegration(filename, 'integrator', integrator, 'options', odeoptions);
+fprintf('Done, now integrate...\n');
+
+% integrate
+try
+    sol_no_filippov = solveODE(dhandle, timeinterval, initstates, p);
+catch ME
+    warning(initial_filippov_warning_state, 'IFDIFF:chattering');
+    warning('Problem integrating model variant wihtout filippov behaviour...');
+    rethrow(ME);
+end
+fprintf('Done \n');
+
+% results
+T = t0:0.01:tf;
+Y_no_fil = deval(sol_no_filippov, T);
+
+figure(1); hold on; box on;
+
+integrator_steps_marker_vals = zeros(length(sol_no_filippov.x));
+
+plot(T, Y_no_fil(1,:), 'LineWidth', 2);
+plot(T, Y_no_fil(2,:), 'LineWidth', 2);
+plot(sol_no_filippov.x, integrator_steps_marker_vals, 'rx', 'MarkerSize', 8, 'LineWidth', 1, 'Color', [0, 0, 0.5]);
+xline(sol_no_filippov.switches, '--r', 'LineWidth', 1.5);
+
+xlabel('Time (s)', 'FontSize', 12);
+ylabel('States', 'FontSize', 12);
+title('ODE Solution to Friction Model', 'FontSize', 14);
+
+legend({'Position of Weight','Velocity of Weight'}, 'Location', 'best');
+grid on;
+
+set(gca, 'FontSize', 12, 'LineWidth', 1.2);
+
+%% Computing VDE-sensitivities
+
+dim_y = size(sol_no_filippov.y, 1);
+dim_p = length(p);
+FDstep = generateFDstep(dim_y, dim_p);
+integrator_options = odeset('AbsTol', 1e-14, 'RelTol', 1e-12);
+try
+    sensitivities_function_VDE = generateSensitivityFunction(dhandle, sol_no_filippov, FDstep, 'integrator_options', integrator_options);
+    sens = sensitivities_function_VDE(T);
+catch ME
+    warning(initial_filippov_warning_state, 'IFDIFF:chattering');
+    warning('Problem with sensitivities in model variant wihtout filippov behaviour...');
+    rethrow(ME)
+end
+Gy11 = arrayfun(@(x) x.Gy(1, 1), sens);
+Gy12 = arrayfun(@(x) x.Gy(1, 2), sens);
+Gy21 = arrayfun(@(x) x.Gy(2, 1), sens);
+Gy22 = arrayfun(@(x) x.Gy(2, 2), sens);
+
+figure(500); box on;
+
+Gy = {Gy11, Gy12, Gy21, Gy22};
+names = {'Gy11','Gy12','Gy21','Gy22'};
+
+% plot the sensitivities w.r.t. the initial states
+for k = 1:4
+    subplot(2,2,k)
+    hold on
+
+    scatter(T, Gy{k}, 'LineWidth',0.5,'Color',[0 0.5 0],'Marker','.')
+    plot(sol_no_filippov.x, integrator_steps_marker_vals, 'rx','MarkerSize',8,'LineWidth',1,'Color',[0 0 0.5])
+
+    xline(sol_no_filippov.switches,'--r','LineWidth',1.5);
+
+    xlabel('Time (s)','FontSize',12)
+    ylabel(names{k},'FontSize',12)
+    title(names{k},'FontSize',14)
+    legend(names{k},'Location','best')
+    grid on
+    set(gca,'FontSize',12,'LineWidth',1.2)
+end
+
+
+%% Solving the friction model variant with filippov mode
+% preprocessing
+fprintf('Preprocessing...\n  ');
+filenamefil = 'friction_RHS_filippov';
+dhandle_filippov = prepareDatahandleForIntegration(filenamefil, 'integrator', integrator, 'options', odeoptions);
+fprintf('Done, now integrate...\n');
+
+% integrate
+try
+    sol_filippov = solveODE(dhandle_filippov, timeinterval, initstates, p);
+catch
+    warning(initial_filippov_warning_state, 'IFDIFF:chattering');
+    warning('Problem integrating model variant wiht filippov behaviour...');
+    rethrow(ME);
+end
+fprintf('Done \n');
+
+% results
+T = t0:0.01:tf;
+Y_filippov = deval(sol_filippov, T);
+
+figure(2); hold on; box on;
+
+integrator_steps_marker_fil = zeros(length(sol_filippov.x));
+
+plot(T, Y_filippov(1,:), 'LineWidth', 2);
+plot(T, Y_filippov(2,:), 'LineWidth', 2);
+plot(sol_filippov.x, integrator_steps_marker_fil, 'rx', 'MarkerSize', 8, 'LineWidth', 1, 'Color', [0, 0, 0.5]);
+
+xline(sol_filippov.switches, '--r', 'LineWidth', 1.5);
+
+xlabel('Time (s)', 'FontSize', 12);
+ylabel('States', 'FontSize', 12);
+title('ODE Solution to Friction Model (Filippov)', 'FontSize', 14);
+
+legend({'Position of Weight','Velocity of Weight'}, 'Location', 'best');
+grid on;
+
+set(gca, 'FontSize', 12, 'LineWidth', 1.2);
+
+%% Compute filippov sensitivities
+% to do
+
+%% creating difference plot
+
+figure(3); hold on; box on;
+
+integrator_steps_marker_fil = epsilon * ones(length(sol_filippov.x));
+fildiff = abs(Y_filippov - Y_no_fil);
+
+plot(T, fildiff(1,:), T, fildiff(2,:), 'LineWidth', 2);
+plot(sol_filippov.x, integrator_steps_marker_fil, 'rx', 'MarkerSize', 8, 'LineWidth', 1, 'Color', [0, 0, 0.5]);
+
+xline(sol_no_filippov.switches, '--r', 'LineWidth', 1.5);
+
+xlabel('Time (s)', 'FontSize', 12);
+ylabel('Difference', 'FontSize', 12);
+title('Difference of filippov and non-filippov integrations', 'FontSize', 14);
+
+legend({'Diff Position','Diff Velocity'}, 'Location', 'best');
+grid on;
+
+set(gca, 'FontSize', 12, 'LineWidth', 1.2, 'YScale', 'log');
+
+%% Display of the switching functions
+
+nu_rel = (Y_no_fil(2,:) - p(3));
+
+figure(4); 
+subplot(2, 1, 1); hold on; box on;
+
+scatter(T, nu_rel, 'LineWidth', 0.5, Marker='.');
+plot(sol_filippov.x, 0, 'rx', 'MarkerSize', 8, 'LineWidth', 1, 'Color', [0, 0, 0.5]);
+
+xline(sol_no_filippov.switches, '--r', 'LineWidth', 1.5);
+
+xlabel('Time (s)', 'FontSize', 12);
+ylabel('\nu_{rel} (switching function)', 'FontSize', 12);
+title('Evolution of first switching condition', 'FontSize', 14);
+
+legend({'\nu_{rel}'}, 'Location', 'best');
+grid on;
+
+set(gca, 'FontSize', 12, 'LineWidth', 1.2);
+
+subplot(2, 1, 2); hold on; box on;
+
+scatter(T, nu_rel, 'LineWidth', 0.5, Marker='.');
+plot(sol_filippov.x, 0, 'rx', 'MarkerSize', 8, 'LineWidth', 1, 'Color', [0, 0, 0.5]);
+
+xline(sol_no_filippov.switches, '--r', 'LineWidth', 1.5);
+
+xlabel('Time (s)', 'FontSize', 12);
+ylabel('\nu_{rel} (switching function)', 'FontSize', 12);
+title('Evolution of first switching condition', 'FontSize', 14);
+ylim([-2 * epsilon 2 * epsilon]);
+legend({'\nu_{rel}'}, 'Location', 'best');
+grid on;
+set(gca, 'FontSize', 12, 'LineWidth', 1.2);
+
+% return warining state to what it was before running the code
+warning(initial_filippov_warning_state, 'IFDIFF:chattering');

toolbox/examples/friction_model/helpers/F_fric_fil.m

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+function res = F_fric_fil(x, mu_rel, epsilon, F_s, k, delta)
+if mu_rel <= 0
+    res = F_s / (1 - delta * mu_rel);
+else
+    res = -F_s / (1 + delta * mu_rel);
+end
+end
+

toolbox/examples/friction_model/helpers/F_fric_no_fil.m

@@ -0,0 +1,20 @@
+function res = F_fric_no_fil(x, mu_rel, epsilon, F_s, k, delta)
+if mu_rel <= -epsilon
+    res = F_s / (1 - delta * mu_rel);
+else
+    if mu_rel >= epsilon
+        res = (-F_s) / (1 + delta * mu_rel);
+    else
+        if k*x(1) <= -F_s
+            res = F_s;
+        else
+            if k*x(1) >= F_s
+                res = F_s;
+            else
+                res = k*x(1);
+            end
+        end
+    end
+end
+end
+