MATLAB: How to solve nonlinear coupled dgl second order

differential equations second ordernonlinear

Hello everyone,

Does anybody know how to solve the following two differential equotations using ODE45?

My Problem is, that i don't know how to rewrite the phi'' and x'' during the transformation in a system of dgl's first order.

Thanks

Best Answer

Here is an example that I have obtained using ode45; the solution seems quite logical since we a dealing with sine waves.

clear variables
close all
N = 500;
L = 5; g = 9.81; mw = 2; mk = 5; Fan = 4; FR = 2;
q1 = -1/L;
q2 = -g/L;
F1 = -(mk*L)/(mw + mk);
F2 = -F1;
S = (Fan - FR)/(mw + mk);
F = @(t,y) [ y(2) ; 
    (1./(1-q1*F1*(cos(y(1)).^2))).*(0.5*q1*F2*sin(2*y(1)).*y(2).^2+...
        q1*S*cos(y(1)) + q2*sin(y(1)))];
 t0 = -2*pi;
tf = 2*pi;
tspan = t0:(tf-t0)/(N-1):tf;
ic = [0 0];
[t,y] = ode45(F, tspan, ic);
figure
plot(t,y(:,1),'-o')
hold on
plot(t,y(:,2),'-o')
a = title('\theta vs \theta_{prime}');
legend('\theta','\theta_{prime}');
set(a,'fontsize',14);
a = ylabel('y');
set(a,'Fontsize',14);
a = xlabel('t [-2\pi 2\pi]');
set(a,'Fontsize',14);
xlim([t0 tf])
grid
grid minor;

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MATLAB: Solving system of second order differential equations

The return type of the parameter from the function block must be a column vector. So, you can add the below statement at the end, inside the function block:

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I think you were missing this.

MATLAB: How to write a sigma calculation

In part two of the problem, you will have to use the definition of the DTFT to compute X(omega). The resultant function is just a periodic version of X(jw) from part one of the problem. See the following code below.

%%Part 1
clear variables
close all
N= 200;
T = 2;
t  = -2:4/(N-1):2;
x = rectpuls(t,T);
f= -2:4/(N-1):2;
X = T*sin(pi*f*T)./(pi*f*T);
figure
subplot(121)
plot(t,x,'linewidth',2,'color','b')
grid;
a = title('x(t)');
set(a,'fontsize',14);
a = ylabel('x');
set(a,'Fontsize',14);
a = xlabel('t');
set(a,'Fontsize',14);
subplot(122)
plot(f,abs(X)/(max(X)),'linewidth',2,'color','m')
a = title('|X(jw)|');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid
%%Part 2
M = 1000;
N = 7;
n = 0:N-1;
xn = ones(1,N);
w = 8*pi;
omega = -w:2*w/(M-1):w; 
Xn= exp(-1i.*omega.*(N-1)./2).*(sin(omega*N/2)./sin(omega/2)); %define DTFT function
Mag = abs(Xn)/max(Xn);   %compute magnitude
Phase = angle(Xn);       %compute phase
figure
subplot(3,1,1)
stem(n,xn,'linewidth',2,'color','b'); 
a = title('x(n)');
set(a,'fontsize',14);
a = ylabel('xn');
set(a,'Fontsize',14);
a = xlabel('n');
set(a,'Fontsize',14);
grid
subplot(3,1,2)
plot(omega/pi,real(Mag),'linewidth',2,'color','k'); 
a = title('|X(\omega)|');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid
subplot(3,1,3)
plot(omega./pi,Phase,'linewidth',2,'color','k'); 
a = title('<X(\omega)');
set(a,'fontsize',14);
a = ylabel('X');
set(a,'Fontsize',14);
a = xlabel('f');
set(a,'Fontsize',14);
grid

Best Answer

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MATLAB: Solving system of second order differential equations

MATLAB: How to write a sigma calculation

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