Question

We consider 2 coupled harmonic oscillators, as shown in the diagram below.

m/ F(0) win 01 x1(0) 02 x200)

The mass m1 is subjected to an external force F (t).

1. Construct the system of differential equations whose unknowns are the displacements x1 (t) and x2 (t) of each of the 2 masses. (5 points).

2. Solve x₁(t) and x₂(t) in the case where m₁ = 1kg; m₂ = 2kg; k = 1 N / m; F(t) = 0 and x₁(0) = 0; ?₁′(0) = 0; x₂(0) = 0; ?₂′(0) = 0 (10 points).

3. Solve x₁(t) and x₂(t) in the case where m₁ = 1kg; m₂ = 2kg; k = 1 N / m; F(t) = cos(t) and x₁(0) = 0; ?₁′(0) = 0; x₂(0) = 0; ?₂′(0) = 0 (10 points).

Answer 1

1.

clc
clear
close
m1=1; m2=2; k=1; F=0;
tspan=[0 10];
x0=[0;0;0;0];
[t,a]=ode45(@(t,a)[a(2);(F-k*(a(1)-a(3)))/m1; a(4); k*(a(1)-a(3))/m2],tspan,x0);
figure(1)
plot(t,a(:,1))
xlabel('time [sec]'), ylabel('x_1 [m]')
figure(2)
plot(t,a(:,3))
xlabel('time [sec]'), ylabel('x_2 [m]')

X 2 - A Figure 1 Eile Edit View Insert Tools Desktop Window Help R 4 - la VE - 5 junk3.mx + 1 - clc clear 3 - close ml=l; m2=2; k=1; F=0; tspan=[0 10]; 6 - x0=[0;0;0;0]; 7 - [t, a)=ode 45 (@(t,a) [a (2); (F-k* (a (1)-a (3) ) ) /ml; a (4); k* (a (1)-a (3)) /m2), tspan, x0); figure (1) plot(t,a(:,1)) 10 - xlabel('time [sec]'), ylabel('x_1 [m]') A Figure 2 х 11 - figure (2) 12 - plot(t, a(:,3)) Eile Edit View Insert Tools Desktop Window Help > 13 - xlabel('time [sec]'), ylabel('x_2 [m]') La 1 8 - 9 0.5 X, [m] 0 1 -0.5 0.5 -1 0 0 2 *2 In 10 8 4 6 time [sec] -0.5 -1 0 2 8 10 4 6 time [sec]

2.

clc
clear
close
m1=1; m2=2; k=1;
tspan=[0 70];
x0=[0;0;0;0];
[t,a]=ode45(@(t,a)[a(2);(cos(t)-k*(a(1)-a(3)))/m1; a(4); k*(a(1)-a(3))/m2],tspan,x0);
figure(1)
plot(t,a(:,1))
xlabel('time [sec]'), ylabel('x_1 [m]')
figure(2)
plot(t,a(:,3))
xlabel('time [sec]'), ylabel('x_2 [m]')

fx >> A Figure 1 ох File Edit View Insert Tools Desktop Window Help TE 3 - 3 junk3.mx + 1 clc 2 - clear close ml=l; m2=2; k=l; 5 - tspan=[0 70]; x0=[0;0;0;0); 7 - [t, a)=ode 45 (@(t, a) [a (2); (cos(t)-k* (a (1)-a (3)))/ml; a (4); k* (a (1)-a (3))/m2], tspan, x0); figure (1) plot(t, a(:,1)) Figure 2 10 - xlabel('time [sec]'), ylabel('x_1 [m]') 11 - figure (2) File Edit View Insert Tools Desktop Window Help 12 - plot(t,a(:,3)) 13 - xlabel('time [sec]'), ylabel ('x_2 [m]') 2 - 8 2 9 [u] " + D -1 -2 0 10 20 50 60 70 X2 [m] 30 40 time (sec] 0 -2 20 60 80 40 time [sec]