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Oct 30, 2023

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%%%%%%%%% MAE 147 S23 % A.S. Voloshina % Apr 2, 2023 % HW 1 clear variables %% Define constants used in simulation % Mass-spring-damper system constants g = 9.81; % gravity m = 2; % mass (kg) l = 1; % spring rest length c = 0; % damping coefficient k1 = 10; % spring constants for nonlinear spring k2 = 15; % Integration constants endTime = 5; % stop integration after this number of seconds % Initial conditions for your ODE solver y_0 = 0; % initial position of mass relative to y_t yDot_0 = 0; % initial y_Dot %% Integration of differential equations %{ ode45 integrates equations defined in the function systemODEs The inputs to ode45 are as follows: @(t,states) - means integrate the vector states, over time systemODEs(inputs) - means integrate equations found in the function systemODEs, which requires these particular inputs timespan - integrate for this time duration (i.e., end integration when you reach the end of the timespan vector) [y_0 yDot_0] - these are the initial conditions for our integration. We are building the [states] vector, and this is our first input to the integration. In other words, for t=0, we have our initial conditions. Then, for one time tick, ode45 integrates these values and finds the vector [states(1,:)] for t=0+deltaT. Then, [states(1,:)] becomes the initial condition for the next time tick, t = 2*deltaT. ode45 then integrates that and finds [states(2,:)], and so on, until it determines the full [states] vector. Outputs: - t (time of integration, should be the same as the previously defined timespan) - states (the solutions to our ODEs from the integration) %} initialConditions = [y_0 yDot_0]; [t, states] = ode45(@(t,states) systemODEs(t,states, m, g, c, k1, k2),... 0:0.01:endTime, initialConditions); y_t = states(:,1); y_t_Dot = states(:,2);

figure(1); plot(t,y_t); xlabel('time') ylabel('y_t (relative to spring at rest)') %% Plot y_t, y_t_Dot, y_t_DDot on one graph % Part B %Differentiation of y_t_Dot at time = 0.01 initial conditions above y_t_DDot = diff(y_t_Dot)/0.01; figure(2) plot(t,y_t,t,y_t_Dot,t(1:length(y_t_DDot),:),y_t_DDot) xlabel('time') ylabel('y_t (relative to spring at rest)') % You may uncomment the legend below to add a legend to your figures legend('Pos','Vel','Acc') %% Find y_r % Part C % Set c=high and note at what length mass comes to a rest % Or, use roots to solve a 3rd order polynomial c = 15; initialConditions = [y_0 yDot_0]; [t, states] = ode45(@(t,states) systemODEs(t,states, m, g, c, k1, k2),... 0:0.01:endTime, initialConditions); y_t = states(:,1); y_t_Dot = states(:,2); figure(3); plot(t,y_t); xlabel('time') ylabel('y_t (relative to spring at rest)') % From Graph y_r = 0.8932; %% Reset to nominal after Part C initialConditions = [y_0 yDot_0]; [t, states] = ode45(@(t,states) systemODEs(t,states, m, g, c, k1, k2),... 0:0.01:endTime, initialConditions); y_t = states(:,1); y_t_Dot = states(:,2); y_t_DDot = diff(y_t_Dot)/0.01;

%% Plot F_s and F_sl % Part D F_s = k1 .* y_t + k2 .* y_t.^3; %From part c y_r = 0.8932; F_sl = k1 * y_r + k2 * y_r^3 + (k1 + 3 * k2* y_r^2) * (y_t - y_r); y_t_DDot = -F_sl / m - c / m * y_t_Dot + g; figure(4) plot(t,F_s,t,F_sl) xlabel('time') ylabel('Spring Force') legend('Fs','Fsl') figure(5) plot(y_t,F_s,y_t,F_sl) xlabel('y_t') ylabel('Spring Force') legend('Fs','Fsl') %% Part E y_r = 0.8932; F_sl = k1*y_r + k2*y_r^3 + (k1 + 3*k2*y_r^2)*(y_t-y_r); y_t_DDot= -F_sl/m - c/m*y_t_Dot + g; figure(6) plot(t,y_t); hold on plot(t,y_t_Dot); plot(t,y_t_DDot); hold off xlabel('time') ylabel('y_t (relative to spring at rest)') legend('Pos','Vel','Acc') %% Function defining all differential equations function stateDot = systemODEs(~,states, m, g, c, k1, k2) %{ System differential equations. These are the equations of motion that are integrated by ode45. The input to the function is time, states vector, mass and spring parameters. ode45 integrates the vector stateDot, which creates a new states vector for the next time step, and repeats until endTime %}

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% Extract values from the states vector y_t = states(1); y_t_Dot = states(2); % Nonlinear spring Equations of Motion F_s = k1*y_t + k2*y_t^3; y_t_DDot = -F_s/m - c/m*y_t_Dot + g; % Linear spring Equations of Motion % YOUR CODE HERE % Either remove or comment out the code for the Nonlinear spring EoMs % F_sl = ??? % y_t_DDot = ???; % Collect all states stateDot = [y_t_Dot y_t_DDot]'; end