Question

9imereal Condltion and note the behavior of the system. xercise 5. For this exercise you will approximate the solution of an undamped periodically forced mass-spring system with mass m and spring constant k = 1.The second order equation for this system is given by x(t)" + x(t) = Cos(at). Use the initial conditions x(0) = 0 and x'(0) =y(0)-0. Use the improved Euler's method and Excel to generate plots of x(t) versus time for ω 1.1 and ω 1 . Use a time interval of 50 seconds for ω 1 and 150 seconds for w 1.1. For both cases set At-0.1. Discuss and explain your results. hint: look at the envelopes.

using improved eulers method using excel

Answer 1

Matlab code for Euler and Heun's method for 2d clear all close all 888888움움88,,88움움8888888888움움8888888888888888888888888888888888888888 functions for improved Euler method for w=1 f-e (t,x,Y) Y g-e(t,x,y) -xtcos(w %step size %Initial guess iteration for x and y using improved Euler method x-huen ( 1 )=x-0 ; y-huen ( 1 ) =y-0 ; t_huen (1)-0 t_end-50; n (t_end-t_huen)/h; for i-l:n t_huen (1+1 )=t_huen ( 1 )th; ml-f (t_huen(i),x_huen(i),y_ huen(i)); m2-g (t_huen(i),x_huen(i),y huen (i)); pl-f (t_huen (i+1),x_huen (i)+h*ml,y huen (i)+h*m2); p2-g (t_huen (i+1),x_huen(i)+h*ml,y_huen (i)+h*m2); x_huen (i+1)x_huen (i)+(h/2)(ml+pl); y huen(i+1)-y_huen (i)+(h/2) * (m2+p2); figure(1) plot(t huen,x huen) xlabel(time in sec ylabel 'x(t) ') title 'Improved Euler solution plot for w1') clear x huen; clear y huen; clear t huen 응88888888888888888888888888888888888888钐8888888888888888888888 88 %functions for improved Euler method for w-1 f-e (t,x,Y) Y g= @ ( t , x , y ) -x+cos (ytt): %step size h=0.1;

%initial guess x_0=0; yー0-0 ; iteration for x and y using improved Euler method s_huen (1)-x-0 ; y-hue n (1)-y-0; t_huen (1) 0 t_end-150; n-(t_end-t_huen)/h; for i=1:n t huen (i+1)-t huen (i)+h; ml-f (t_huen(i),x_huen(i) ,y_huen(i)); m2-g (t_huen(i),x_huen(i),y huen(i)) p2-g(t_huen(i+1),x_huen(i)+h*ml,y_huen (i)+h*m2); x_huen (i+1)-x_huen (i)+(h/2)*(ml+pl)i y huen (i+1)-y_huen (i)+(h/2) (m2+p2); end figure (2) plot (t_huen, x_huen) xlabel('time in sec.') ylabel'x(t) ) title( 'Improved Euler solution plot for w-l.1') clear x_huen; clear y_huen; clear t_huen fprintf(tFor w-l the period of envelop is higher than that of w-1.1 For w1 the period of envelop is higher than that of w-1.1

Improved Euler solution plot for w 1 25 15 10 10 15 10 20 30 time in sec. 50 Improved Euler solution plot for w-1.1 10 10 50 150 time in sec.

%Matlab code for Euler and Heun's method for 2d
clear all
close all

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%functions for improved Euler method for w=1
w=1;
f=@(t,x,y) y;
g=@(t,x,y) -x+cos(w*t);

%step size
h=0.1;

%Initial guess
x_0=0; y_0=0;

%iteration for x and y using improved Euler method
x_huen(1)=x_0; y_huen(1)=y_0;
t_huen(1)=0; t_end=50;
n=(t_end-t_huen)/h;
for i=1:n
  
    t_huen(i+1)=t_huen(i)+h;
    m1=f(t_huen(i),x_huen(i),y_huen(i));
    m2=g(t_huen(i),x_huen(i),y_huen(i));
  
    p1=f(t_huen(i+1),x_huen(i)+h*m1,y_huen(i)+h*m2);
    p2=g(t_huen(i+1),x_huen(i)+h*m1,y_huen(i)+h*m2);
  
    x_huen(i+1)=x_huen(i)+(h/2)*(m1+p1);
    y_huen(i+1)=y_huen(i)+(h/2)*(m2+p2);
  
end

figure(1)
plot(t_huen,x_huen)
xlabel('time in sec.')
ylabel('x(t)')
title('Improved Euler solution plot for w=1')

clear x_huen; clear y_huen; clear t_huen

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%functions for improved Euler method for w=1
w=1.1;
f=@(t,x,y) y;
g=@(t,x,y) -x+cos(w*t);

%step size
h=0.1;

%Initial guess
x_0=0; y_0=0;

%iteration for x and y using improved Euler method
x_huen(1)=x_0; y_huen(1)=y_0;
t_huen(1)=0; t_end=150;
n=(t_end-t_huen)/h;
for i=1:n
  
    t_huen(i+1)=t_huen(i)+h;
    m1=f(t_huen(i),x_huen(i),y_huen(i));
    m2=g(t_huen(i),x_huen(i),y_huen(i));
  
    p1=f(t_huen(i+1),x_huen(i)+h*m1,y_huen(i)+h*m2);
    p2=g(t_huen(i+1),x_huen(i)+h*m1,y_huen(i)+h*m2);
  
    x_huen(i+1)=x_huen(i)+(h/2)*(m1+p1);
    y_huen(i+1)=y_huen(i)+(h/2)*(m2+p2);
  
end

figure(2)
plot(t_huen,x_huen)
xlabel('time in sec.')
ylabel('x(t)')
title('Improved Euler solution plot for w=1.1')

clear x_huen; clear y_huen; clear t_huen

fprintf('\tFor w=1 the period of envelop is higher than that of w=1.1 \n')

%%%%%%%%%%%%%%%%%%%%%%%% End of code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%