Question

help me with this. Im done with task 1 and on the way to do task 2. but I don't know how to do it. I attach 2 file function of rksys and ode45 ( the first is rksys and second is ode 45) . thank for your help

Consider the spring-mass damper that can be used to model many dynamic systems -- ----- ------- m Applying Newton's Second Law to a free-body diagram of the mass m yields the following ODE: dar m 2 + + +kx = F(t) (1) Where F(t) is a forcing function. Consider the case where the forcing function itself is a damped oscillation: F(t) = Ae-Btcos (wt) (2) For this activity we'll see how we can formulate this ODE for a solution using MATLAB that could be used to study the sensitivity of the system to changes in any of the parameter values. Task 1: Define two new variables Y1 = x and y2 = dt. The resulting relationship between yıand y2 gives us the first equation in the system of first order ODEs that is equivalent to equation (1): dyı dt 592 Note this equation will ALWAYS be the first ODE in the system of first order ODEs written to be equivalent to ANY higher order ODE. Next use the definitions of Yiand y2 to substitute for any terms involving x in equation (1) such that the only derivative term in the equation is not solve the equation for it. This is the second equation in the system. ENGR&240 Applied Numerical Methods TCC Workspace for Task 1 4.64 iz te coscow) Shayag har en Task 2: Write a MATLAB function for those two expressions that you have derived above. Save that function with a file name forcedvibration.m. Use that function, the rksys.m, ode45.m, and ode23s.m to solve the forced vibration problem in Task 1 assuming, A = 1000, B = 100, k = 50, w = n/2, and m = 10. Also assume that x(0) = x(0) = 0. Write your solutions of this problem on a MATLAB script. Plot the displacement (x) and the slope (dx/dt) results for those three methods (i.e. you will have two plots, one for the displacement and the other one for the slope). Label your script as Lastname ForcedVibrationSolution.m. Task 3: Compare the number of data (e.g. t or time stepping) generated by those three method on Task 2. Why are they different? Task 4: Change the B value into 0.1, 1, 10, and 1000. Would you be able to explain the vibration behavior that you observed? Make sure your h for the rksys.m method is sufficient.

Answer 1

%i have created the script as well as the function. You need to change parameters as needed. If any error is there, please tell in comments, I will fix it

y0=[0;0];
tstart=0;
tend=1;%edit it accordingly
h=.1;%change as needed
tspan=[tstart,tend];
[t_23s,y_23s]=ode23s(@forcedvibration,tspan,y0);%calling ode23s
[t_45,y_45]=ode45(@forcedvibration,tspan,y0);%calling ode45
[tp,yp]=rk4sys(@forcedvibration,tspan,y0,h);%calling rk4sys
plot(t_23s,y_23s(:,1),t_45,y_45(:,1),tp,yp(:,1))
xlabel('t')
ylabel('x')
title('displacement')
legend('ode 23s','ode45','rk4sys')
figure
plot(t_23s,y_23s(:,2),t_45,y_45(:,2),tp,yp(:,2))
xlabel('t')
legend('ode 23s','ode45','rk4sys')
ylabel('x''')
title('slope')

%the required function
function y = forcedvibration(t,x)
A=1000;
B=100;
k=50;
w=pi/2;
m=10;
f=A*exp(-B*t).*cos(w.*t);
y=[x(2);(f-k*x(1)-B*x(2))/m];
end