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Two second order differential equation

  1. Oct 12, 2009 #1
    hi, everybody
    i got for homework a pendulum on a cart.

    i solved the system and got two equations.

    (m+M)x'' + M L theta'' = F

    x'' + L theta'' + g* theta = 0

    F = 1000 N, m = 500 kg, M = 1250 kg, l = 10m

    i know how to put them in state space and solve them with SS block in
    simulink.but i don't know how to solve the system with ode23. there are
    no initial conditions stated in a task, i guess there are zero.

    i understand this sintax
    function xdot = vdpol(t,x)
    xdot = [x(1).*(1-x(2).^2)-x(2); x(1)]

    t0 = 0; tf = 20;
    x0 = [0 0.25]'; % Initial conditions
    [t,x] = ode23('vdpol',t0,tf,x0);
    plot(t,x)

    but dont know how to use it on those eq.
     
  2. jcsd
  3. Oct 13, 2009 #2
    Do you know how to solve higher order differential equation using Runge-Kutta method?
    Change your system of equations to the following form first before you use ode23

    [tex]\frac{d\vec{y}}{dt} = \vec{f}(t,\vec{y})[/tex]

    where
    [tex]\vec{y} = ( x , \dot{x} , \theta , \dot{\theta})^t[/tex]
     
  4. Oct 14, 2009 #3
    thank you for your reply.
    i dont know if that is the method i am using but i know how to transform
    this second order to first order eq.
    and i got.
    [tex]\ddot{\theta} +\frac{F}{m*l}+ \frac{(m+M)*g}{l*m}\theta=0[/tex]
    [tex]\ddot{x}-\frac{F}{m}-\frac{g*M}{m}\theta=0[/tex]

    [tex]x_{1} = \dot{x} -> \dot{x}_{1}= \ddot{x}[/tex]
    [tex]x_{2} = x -> \dot{x}_{2} = \dot{x} = x_{1} [/tex]
    [tex]\theta_{1} = \dot{\theta} -> \dot{\theta}_{1}= \ddot{\theta}[/tex]
    [tex]x_{2} = \theta -> \dot{\theta}_{2} = \dot{\theta} = \theta_{1} [/tex]

    and now i transform in
    [tex]\dot{\theta}_{1} = -\frac{F}{m*l}-\frac{(m+M)*g}{l*m}\theta_{2}[/tex]
    [tex]\dot{\theta}_{2} = \theta_{1}[/tex]
    [tex]\dot{x}_{1} = \frac{F}{m}+\frac{g*M}{m}\theta_{2}[/tex]
    [tex]\dot{x}_{2} = x_{1}[/tex]

    and now i know how to put that in matlab
    function thetadot = thetam(t, theta)
    thetadot = [-0.2 - theta(2).*3.4335; theta(1)] % here are numbers because matlab
    % cant "see" the variables F,m,M...
    t0 = 0;
    tf = 10;
    theta0 = [0 0]';
    [t,theta] = ode23('thetam',t0,tf,theta0);
    plot(t,theta)
    plot(t, theta(84:1:166))

    and i get two graphs. one of them is the on i need.
    can you explain me what is the second one, please.

    but i dont know how to write in matlab solution for x
    becase it has a theta in it eq??
     
  5. Oct 14, 2009 #4
    try checking the size for theta. I think it's size is n by 2. That's why you have two graphs I think. It is just like using the command plot(t,theta(:,1),t,theta(:,2)).

    The equation that you solve with matlab actually can be solve analytically because it is an inhomogeneous linear DE with constant coefficients.
     
  6. Oct 14, 2009 #5
    yes, i know it has two columns. what is in a first? also solution to the eq?
    problem is i dont know how to solve it for x... can you say exactly how
    because i have no clue
     
  7. Oct 14, 2009 #6
    My guess is that your second graph is t against theta. You must have known better. You wrote the program.

    Try using the substitution that I wrote in my first post.

    [tex] y_1= x , y_2=\dot{x},y_3=\theta , y_4=\dot{\theta}[/tex]
     
  8. Oct 14, 2009 #7
    i am sorry, but i cant see how is your's supstitution different form
    mine when i wrote what i did to that point
     
  9. Oct 14, 2009 #8
    Not much different actually :biggrin: . But you use subscripts for both x and theta that confuse me. It looks like you want to solve the two equations separately. What I have in mind is solving the two equations simultaneously using the four variables y1, y2, y3, and y4.

    Then the plots for your solution are plot(t,y(:,1)) for x and plot(t,y(:,3)) for theta.
     
  10. Oct 21, 2009 #9
    i forgot to write the solution... if anyone was interested or have similar problem
    [tex]y_{1} = \dot{x} [/tex]
    [tex]y_{2} = x [/tex]
    [tex]y_{3} = \dot{\theta}[/tex]
    [tex]y_{4} = \theta[/tex]
    [tex]\dot{y}_{1} = \ddot{x} [/tex]
    [tex]\dot{y}_{2} = \dot{x} = y_{1} [/tex]
    [tex]\dot{y}_{3} = \ddot{\theta} [/tex]
    [tex]\dot{y}_{4} = \dot{\theta} = y_{3} [/tex]

    [tex] \dot{y}_{1} = \frac{F}{m} + \frac{gM}{m}y_{4} [/tex]
    [tex] \dot{y}_{3} = -\frac{F}{ml} - \frac{(m+M)g}{ml}y_{4} [/tex]
    [tex] \dot{y}_{4} = y_{3} [/tex]
    [tex] \dot{y}_{2} = y_{1} [/tex]

    and when we put it in matlab
    _________________________
    function ydot = ydot(t, y)
    ydot = [2 + y(4).*24.525; y(1); -0.2 - y(4).*3.4335; y(3)]
    %this is ydot.m file
    _________________________
    M = 1250;
    m=500;
    l = 10;
    g = 9.81;
    F = 1000;
    t0 = 0;
    tf = 10;
    poc0 = [0 0 0 0]';
    [t,rj]= ode23('xm', t0, tf, poc0)
    plot(t,rj)

    problem is similar to http://www.myphysicslab.com/pendulum_cart.html#indirect", except force is constant in my problem.
     
    Last edited by a moderator: Apr 24, 2017
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